Bevezet\303\251s a matematik\303\241ba J\303\241rai Antal Ezek a programok csak szeml\303\251ltet\303\251sre szolg\303\241lnak.
<Text-field style="Heading 1" layout="Heading 1">1. Halmazok</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">2. Term\303\251szetes sz\303\241mok</Font></Text-field>
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">3. A sz\303\241mfogalom b\305\221v\303\255t\303\251se</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">4. V\303\251ges halmazok</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">5. V\303\251gtelen halmazok</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">6. Sz\303\241melm\303\251let</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">6.1. Oszthat\303\263s\303\241g</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.1. Oszthat\303\263s\303\241g a term\303\251szetes sz\303\241mok k\303\266r\303\251ben.</Font></Text-field> restart; with(numtheory); 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 divisors(60); PC4iIiIiIiMiIiQiIiUiIiYiIiciIzUiIzciIzoiIz8iI0kiI2c= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->6.1.2. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.1.3. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.1.4. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.5. Az o<Font encoding="UTF-8">szthat\303\263s\303\241g </Font>tulajdons<Font encoding="UTF-8">\303\241</Font>gai <Font encoding="UTF-8">a term\303\251szetes sz\303\241mok k\303\266r\303\251ben.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.6. T\303\266rzssz\303\241mok \303\251s pr\303\255msz\303\241mok.</Font></Text-field> isprime(16); isprime(17); divisors(17); nextprime(15); prevprime(18); SSZmYWxzZUclKnByb3RlY3RlZEc= SSV0cnVlRyUqcHJvdGVjdGVkRw== PCQiIiIiIzw= IiM8 IiM8 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->6.1.7. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.8. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.9. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.1.10. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.11. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.12. Oszthat\303\263s\303\241g egys\303\251gelemes integrit\303\241si tartom\303\241nyban.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.13. Az oszthat\303\263s\303\241g tulajdons\303\241gai egys\303\251gelemes integrit\303\241si tartom\303\241nyban.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.14. Asszoci\303\241ltak \303\251s egys\303\251gek.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.15. Felbonthatatlan elem \303\251s pr\303\255melem.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.16. Oszthat\303\263s\303\241g az eg\303\251sz sz\303\241mok k\303\266r\303\251ben.</Font></Text-field> divisors(-60); igcd(12,18); igcd(-12,18); ilcm(12,18); ilcm(-12,-18); PC4iIiIiIiMiIiQiIiUiIiYiIiciIzUiIzciIzoiIz8iI0kiI2c= IiIn IiIn IiNP IiNP LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.17. P\303\251lda: Gauss-eg\303\251szek.</Font></Text-field> with(GaussInt); GInormal(2-3*I); N0BJKEdJYmFzaXNHNiJJKEdJY2hyZW1HRiRJKkdJZGl2aXNvckdGJEkqR0lmYWNwb2x5R0YkSSlHSWZhY3NldEdGJEkpR0lmYWN0b3JHRiRJKkdJZmFjdG9yc0dGJEkmR0lnY2RHRiRJKEdJZ2NkZXhHRiRJKkdJaGVybWl0ZUdGJEkoR0lpc3NxckdGJEkmR0lsY21HRiRJKkdJbWNtYmluZUdGJEkmR0ltb2RHRiRJKkdJbmVhcmVzdEdGJEkoR0lub2RpdkdGJEknR0lub3JtR0YkSSlHSW5vcm1hbEdGJEkoR0lvcmRlckdGJEkmR0lwaGlHRiRJKEdJcHJpbWVHRiRJKkdJcXVhZHJlc0dGJEkmR0lxdW9HRiRJJkdJcmVtR0YkSShHSXJvb3RzR0YkSShHSXNpZXZlR0YkSShHSXNtaXRoR0YkSSpHSXNxcmZyZWVHRiRJJ0dJc3FydEdGJEktR0l1bml0bm9ybWFsR0Yk LCYiIiQiIiIqJiIiI0YkXiNGJEYkRiQ= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.18. Legnagyobb k\303\266z\303\266s oszt\303\263, legkisebb k\303\266z\303\266s t\303\266bbsz\303\266r\303\266s, relativ primek.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.1.19. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.1.20. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.1.21. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.1.22. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.23. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.24. Legnagyobb k\303\266z\303\266s oszt\303\263, legkisebb k\303\266z\303\266s t\303\266bbsz\303\266r\303\266s az eg\303\251sz sz\303\241mok k\303\266r\303\251ben.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.25. B\305\221v\303\255tett euklid\303\251szi algoritmus.</Font></Text-field> exgcd:=proc(a::integer,b::integer) local n,x,y,r,q; x[0]:=1;y[0]:=0;r[0]:=a;x[1]:=0;y[1]:=1;r[1]:=b;n:=0; do if r[n+1]=0 then return x[n],y[n],r[n] fi; q[n+1]:=floor(r[n]/r[n+1]);r[n+2]:=r[n]-q[n+1]*r[n+1]; x[n+2]:=x[n]-q[n+1]*x[n+1];y[n+2]:=y[n]-q[n+1]*y[n+1]; n:=n+1; od; end; Zio2JCdJImFHNiJJKGludGVnZXJHJSpwcm90ZWN0ZWRHJ0kiYkdGJkYnNidJIm5HRiZJInhHRiZJInlHRiZJInJHRiZJInFHRiZGJkYmQyo+JkYtNiMiIiEiIiI+JkYuRjRGNT4mRi9GNEYlPiZGLTYjRjZGNT4mRi5GPUY2PiZGL0Y9Rio+RixGNT8oRiZGNkY2RiZJJXRydWVHRihDKEAkLyZGLzYjLCZGLEY2RjZGNkY1TzYlJkYtNiNGLCZGLkZOJkYvRk4+JkYwRkktSSZmbG9vckdGJjYjKiZGUEY2RkghIiI+JkYvNiMsJkYsRjYiIiNGNiwmRlBGNiomRlJGNkZIRjZGVz4mRi1GWiwmRk1GNiomRlJGNiZGLUZJRjZGVz4mRi5GWiwmRk9GNiomRlJGNiZGLkZJRjZGVz5GLEZKRiZGJkYm exgcd(12,18); exgcd(-12,-18); igcdex(12,18,'x','y'); x; y; NiUhIiIiIiIiIic= NiUhIiIiIiIhIic= IiIn ISIi IiIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.26. Megjegyz\303\251s.</Font></Text-field> exgcd:=proc(a::integer,b::integer) local x0,x1,x2,y0,y1,y2,r0,r1,r2,q; x0:=1;y0:=0;r0:=a;x1:=0;y1:=1;r1:=b; do if r1=0 then return x0,y0,r0 fi; q:=floor(r0/r1);r2:=r0-q*r1; x2:=x0-q*x1;y2:=y0-q*y1; r0:=r1;r1:=r2;x0:=x1;x1:=x2;y0:=y1;y1:=y2; od; end; Zio2JCdJImFHNiJJKGludGVnZXJHJSpwcm90ZWN0ZWRHJ0kiYkdGJkYnNixJI3gwR0YmSSN4MUdGJkkjeDJHRiZJI3kwR0YmSSN5MUdGJkkjeTJHRiZJI3IwR0YmSSNyMUdGJkkjcjJHRiZJInFHRiZGJkYmQyk+RiwiIiI+Ri8iIiE+RjJGJT5GLUY6PkYwRjg+RjNGKj8oRiZGOEY4RiZJJXRydWVHRihDLUAkL0YzRjpPNiVGLEYvRjI+RjUtSSZmbG9vckc2JEYoSShfc3lzbGliR0YmNiMqJkYyRjhGMyEiIj5GNCwmRjJGOComRjVGOEYzRjhGTT5GLiwmRixGOComRjVGOEYtRjhGTT5GMSwmRi9GOComRjVGOEYwRjhGTT5GMkYzPkYzRjQ+RixGLT5GLUYuPkYvRjA+RjBGMUYmRiZGJg== exgcd(12,18); NiUhIiIiIiIiIic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.27. P\303\251lda: b\305\221v\303\255tett euklideszi algoritmus.</Font></Text-field> exgcd(172,62); NiUhIioiI0QiIiM= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.28. K\303\266vetkezm\303\251ny.</Font></Text-field> igcd(12,18,10); IiIj LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.29. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.30. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.31. A sz\303\241melm\303\251let alapt\303\251tele.</Font></Text-field> ifactor(720); KigpLUkhRzYiNiMiIiMiIiUiIiIpLUYlNiMiIiRGKEYqLUYlNiMiIiZGKg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.1.32. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.33. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.34. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.35. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.36. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.37. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.38. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.39. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.40. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.1.41. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.42. Feladat: Lam<Font encoding="UTF-8">\303\251</Font> t<Font encoding="UTF-8">\303\251</Font>tele.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.43. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.44. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.45. Feladat: bin<Font encoding="UTF-8">\303\241</Font>ris lnko.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.46. Euklid\303\251sz t\303\251tele.</Font></Text-field> P:=2; ifactor(P+1); P:=P*3; ifactor(P+1); P:=P*5; ifactor(P+1); P:=P*7; ifactor(P+1); P:=P*11; ifactor(P+1); P:=P*13; ifactor(P+1); IiIj LUkhRzYiNiMiIiQ= IiIn LUkhRzYiNiMiIig= IiNJ LUkhRzYiNiMiI0o= IiQ1Iw== LUkhRzYiNiMiJDYj IiU1Qg== LUkhRzYiNiMiJTZC IiZJKyQ= KiYtSSFHNiI2IyIjZiIiIi1GJDYjIiQ0JkYo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.47. Megjegyz\303\251s.</Font></Text-field> ifactor(P+2); ifactor(P+3); ifactor(P+4); ifactor(P+5); ifactor(P+6); ifactor(P+7); KiYpLUkhRzYiNiMiIiMiIiUiIiItRiU2IyIleD1GKg== KigpLUkhRzYiNiMiIiQiIiMiIiItRiU2IyIjWkYqLUYlNiMiI3JGKg== KiYtSSFHNiI2IyIiIyIiIi1GJDYjIiY8XSJGKA== KiYtSSFHNiI2IyIiJiIiIi1GJDYjIiUyZ0Yo KigpLUkhRzYiNiMiIiNGKCIiIi1GJTYjIiIkRiktRiU2IyIlLkRGKQ== KiYpLUkhRzYiNiMiIigiIiMiIiItRiU2IyIkOCdGKg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.48. Megjegyz\303\251s.</Font></Text-field> N:=6; for n to N do s:=0: for i from 2 to 10^n do if isprime(i) then s:=s+1 fi; od; s,10^n/ln(10.^n); od; IiIn IiIh NiQiIiUkIis+WyVITSUhIio= IiIh NiQiI0QkIis1Q1pyQCEiKQ== IiIh NiQiJG8iJCIrdCNbd1ciISIo IiIh NiQiJUg3JCIrMGl0JjMiISIn IiIh NiQiJSNmKiQiK1UnKillbykhIic= IiIh NiQiJilceSQiK2s4Q1FzISIm N:=25;for n to N do n,pi(10^n),10^n/ln(10.^n) od; IiNE NiUiIiIiIiUkIis+WyVITSUhIio= NiUiIiMiI0QkIis1Q1pyQCEiKQ== NiUiIiQiJG8iJCIrdCNbd1ciISIo NiUiIiUiJUg3JCIrMGl0JjMiISIn NiUiIiYiJSNmKiQiK1UnKillbykhIic= NiUiIiciJilceSQiK2s4Q1FzISIm NiUiIigiJ3pYbSQiKyYpbz8vaSEiJQ== NiUiIikiKGI5dyYkIitENW9HYSEiJA== NiUiIioiKU12JTMmJCIrVlVcRFshIiM= NiUiIzUiKjZEMGIlJCIrPlslSE0lISIi NiUiIzYiKzhbMD1UJCIrYTs4W1IiIiE= NiUiIzciLD0/InpnUCQiKyNvPyI+TyIiIg== NiUiIzgiLVJvYGxnTSQiKyV5RTJNJCIiIw== NiUiIzkiLi0zdlRcPyQkIitVTTUtSiIiJA== NiUiIzoiL3BFVXFYJSlIJCIrWWxIJipHIiIl NiUiIzsiMERSLlQkUSN6IyQiKzcwTTlGIiIm NiUiIzwiMUxVbGRyYkJFJCIrQk1uYUQiIic= NiUiIz0iMmczdShHYSpSWiMkIitBcnU3QyIiKA== NiUiIz4iMzJZTXdzbWRTQiQiK0ovdyZHIyIiKQ== NiUiIz8iNFMpPTRjLSc+M0EjJCIrNUNackAiIio= NiUiI0AiNUc+dD1nW3BzN0AkIitpKm8hbz8iIzU= Warning, computation interrupted LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.1.49. Kanonikus alak.</Text-field> ifactor(-720); ifactors(-720); LCQqKCktSSFHNiI2IyIiIyIiJSIiIiktRiY2IyIiJEYpRistRiY2IyIiJkYrISIi NyQhIiI3JTckIiIjIiIlNyQiIiRGJjckIiImIiIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.50. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.51. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.52. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.53. K\303\266vetkezm\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.54. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.1.55. Erathoszthen\303\251sz szit\303\241ja.</Font></Text-field> N:=1000; B:=Array(1..N,1); sieve:=proc() local p,i; global N,B; B[1]:=0; p:=1; do while B[p]=0 do p:=p+1; od; if p^2>N then return fi; i:=p^2; while i<=N do B[i]:=0; i:=i+p; od; p:=p+1; od; end; sieve(); ArrayElems(B); IiUrNQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JVEiQkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNjBRIzo9RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJWZvcm1HUSZpbmZpeEYnLyUnbHNwYWNlR1EvdGhpY2ttYXRoc3BhY2VGJy8lJ3JzcGFjZUdGTy8lKG1pbnNpemVHUSIxRicvJShtYXhzaXplR1EpaW5maW5pdHlGJy1JKG1hY3Rpb25HRiQ2JC1JKG1mZW5jZWRHRiQ2JS1JJ210YWJsZUdGJDYmLUkkbXRyR0YkNiMtSSRtdGRHRiQ2Iy1GIzYkLUYsNiVRLH4xfi4ufjEwMDB+RidGL0YyLUYsNiVRJkFycmF5RidGL0YyLUZcbzYjLUZfbzYjLUYjNiQtRiw2JVEsRGF0YX5UeXBlOn5GJ0YvRjItRiw2JVEpYW55dGhpbmdGJ0YvRjItRlxvNiMtRl9vNiMtRiM2JC1GLDYlUSpTdG9yYWdlOn5GJ0YvRjItRiw2JVEscmVjdGFuZ3VsYXJGJ0YvRjItRlxvNiMtRl9vNiMtRiM2JC1GLDYlUShPcmRlcjp+RidGL0YyLUYsNiVRLkZvcnRyYW5fb3JkZXJGJ0YvRjIvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRicvJSthY3Rpb250eXBlR1EtYnJvd3NlcnRhYmxlRic= Zio2IjYkSSJwR0YjSSJpR0YjRiNGI0MlPiZJIkJHRiM2IyIiIiIiIT5GJUYsPyhGI0YsRixGI0kldHJ1ZUclKnByb3RlY3RlZEdDJz8oRiNGLEYsRiMvJkYqNiNGJUYtPkYlLCZGJUYsRixGLEAkMkkiTkdGIyokKUYlIiIjRixPRiM+RiZGPD8oRiNGLEYsRiMxRiZGO0MkPiZGKjYjRiZGLT5GJiwmRiZGLEYlRixGN0YjNiRGO0YqRiM= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->6.1.56. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.57. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.1.58. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.1.59. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.1.60. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.1.61. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.62. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.63. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.64. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.65. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.66. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.67. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.68. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.69. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.70. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.71. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.72. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.73. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.74. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.75. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.76. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok megold<Font encoding="UTF-8">\303\241</Font>sokkal.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.1.77. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok.</Text-field>
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">6.2. Kongruenci\303\241k</Font></Text-field> restart;
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.1. Kongruenci\303\241k.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->6.2.2. Feladat. </Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.3. Marad\303\251koszt\303\241lyok.</Font></Text-field> 13 mod 7; modp(13,7); `mod`(13,7); mods(13,7); `mod`:=mods; 13 mod 7; IiIn IiIn IiIn ISIi SSVtb2RzRyUqcHJvdGVjdGVkRw== ISIi `mod`:=modp; [2,10] mod 8; [2,8] mod 8; [8,1,10,19,4,29,-10,7] mod 8; mods(%,8); [1,19,29,7] mod 8; SSVtb2RwRyUqcHJvdGVjdGVkRw== NyQiIiNGIw== NyQiIiMiIiE= NyoiIiEiIiIiIiMiIiQiIiUiIiYiIiciIig= NyoiIiEiIiIiIiMiIiQiIiUhIiQhIiMhIiI= NyYiIiIiIiQiIiYiIig= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.2.4. Komplemens <Font encoding="UTF-8">\303\241</Font>br<Font encoding="UTF-8">\303\241</Font>zol<Font encoding="UTF-8">\303\241</Font>s.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.5. T\303\251tel.</Font></Text-field> 1/3 mod 8; 1/2 mod 8; 1/1 mod 5; 1/2 mod 5; 1/3 mod 5; 1/4 mod 5; IiIk Error, the modular inverse does not exist IiIi IiIk IiIj IiIl
<Text-field style="Heading 3" layout="Heading 3">->6.2.6. Feladat.</Text-field> 1/5 mod 17; 9*11 mod 17; (15+10)/(3+5) mod 17; mul(i,i=1..16) mod 17; IiIo IiM5 IiIi IiM7 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.2.7. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.8. Diszkr<Font encoding="UTF-8">\303\251</Font>t logaritmus probl<Font encoding="UTF-8">\303\251</Font>ma.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.2.9. Gyors hatv\303\241nyoz\303\241s.</Font></Text-field> fastexp:=proc(g,n::posint,mult::procedure) local j,x,b; b:=convert(n,base,2); x:=g; for j from nops(b)-1 to 1 by -1 do x:=mult(x,x); if b[j]=1 then x:=mult(x,g) fi; od; x; end; Zio2JUkiZ0c2IidJIm5HRiVJJ3Bvc2ludEclKnByb3RlY3RlZEcnSSVtdWx0R0YlSSpwcm9jZWR1cmVHRik2JUkiakdGJUkieEdGJUkiYkdGJUYlRiVDJj5GMC1JKGNvbnZlcnRHRik2JUYnSSViYXNlR0YlIiIjPkYvRiQ/KEYuLCYtSSVub3BzR0YpNiNGMCIiIkY+ISIiRj9GPkkldHJ1ZUdGKUMkPkYvLUYrNiRGL0YvQCQvJkYwNiNGLkY+PkYvLUYrNiRGL0YkRi9GJUYlRiU= fastexp(2,11,(x,y)->x*y); IiVbPw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.2.10. Diffie-Hellmann-Merkle-kulcscsere.</Text-field> with(numtheory); 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 q:=safeprime(43598760126543278905668798765412345657890987654234186); p:=(q-1)/2; isprime(p); g:=3; IlZuMEVhdyk0KnlsWEJUbCgpem9jISp5S2FFLHcpZlY= IlYkR0lyI1FcWCpHRzwxRlEqUk1HWFI7RmorUSp6QA== SSV0cnVlRyUqcHJvdGVjdGVkRw== IiIk T:=convert(floor(time()*1000)+3141592 mod (111^3),base,111);while nops(T)<3 do T:=[op(TT),0] od; NyUiJDAiIiQ0IiIjSw== N:=3; T:=vector(N+3,T); n:=0; m:=0; c:="a"; IiIk 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 IiIh IiIh USJhNiI= nexttime:=proc() local i,S; global T,TT,N,m,n,c; S:="0123456789-abcdefghijklmnopqrstuvwxyz"; m:=1+(T[3+n] mod 3); i:=T[m] mod 37; if S[i+1]=c then n:=n+1; T[3+n]:=floor(time()*1000) mod 3*37*16; T[m]:=T[3+n] fi; if n=N then print(T); return fi; m:=1+(T[3+n] mod 3); i:=T[m] mod 37; print(T,S[i+1]); c:=readline(terminal); while true do i:=0 od; end; 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 nexttime(); LUkobWZlbmNlZEc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUknbXRhYmxlR0YkNiMtSSRtdHJHRiQ2KC1JJG10ZEdGJDYjLUkjbW5HRiQ2JFElMTI3MUYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1GMjYjLUY1NiRRJDExMEYnRjgtRjI2Iy1GNTYkUSUxMjQ1RidGOC1GMjYjLUY1NiRRIzIwRidGOEZARjEvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRic= convert(T,list); %[4..N+3]; map(t->t mod 16,%); x:=add(%[i]*16^(i-1),i=1..nops(%)); X:=g&^x mod q; NygiJDAiIiQ0IiIjSyZJIlRHNiI2IyIiJSZGJzYjIiImJkYnNiMiIic= NyUmSSJURzYiNiMiIiUmRiQ2IyIiJiZGJDYjIiIn NyUmSSJURzYiNiMiIiUmRiQ2IyIiJiZGJDYjIiIn LCgmSSJURzYiNiMiIiUiIiIqJiIjO0YoJkYkNiMiIiZGKEYoKiYiJGMjRigmRiQ2IyIiJ0YoRig= LUkjJl5HNiI2JCIiJCwoJkkiVEdGJDYjIiIlIiIiKiYiIztGLCZGKTYjIiImRixGLComIiRjI0YsJkYpNiMiIidGLEYs x:=76813406921544738654231750679890923210164916436739; X:=g&^x mod q; y:=47614360991784651324765987666789098766542333432543; Y:=g&^y mod q; IlNSblY7XDs1SyM0Kil6MXZKVWxRWmFAcFM4byg= IlZvVnU/X0YhRyRIQDp0XTpgZ1BIeDlVXTpEN2wj IlNWRFZMQmFtKCk0Knltdylmd0M4bCV5Iio0Tzl3JQ== IlZFUmcxRik+InBzeDM8NDppcHlrW1YmeVdXaz9F X&^y mod q; Y&^x mod q; IlVZKEdiY25WIjNHJClRMSIpPjo4UEJeISpSKD4qUilcIw== IlVZKEdiY25WIjNHJClRMSIpPjo4UEJeISpSKD4qUilcIw==
<Text-field style="Heading 3" layout="Heading 3">6.2.11. Feladat.</Text-field> [[3^i mod 17,i]$i=1..16]; sort(map(x->[x[1] mod 17,x[2]],%),(x,y)->x[1]<y[1]); NzI3JCIiJCIiIjckIiIqIiIjNyQiI0ZGJDckIiMiKSIiJTckIiRWIyIiJjckIiRIKCIiJzckIiUoPSMiIig3JCIlaGwiIik3JCImJG8+Ric3JCImXCFmIiM1NyQiJ1pyPCIjNjckIidUOWAiIzc3JCIoQlZmIiIjODckIihwSHklIiM5NyQiKTIqW1YiIiM6NyQiKUBuL1YiIzs= NzI3JCIiIiIjOzckIiIjIiM5NyQiIiRGJDckIiIlIiM3NyQiIiZGLzckIiInIiM6NyQiIigiIzY3JCIiKSIjNTckIiIqRic3JEY4Rio3JEY1RjQ3JEYtIiM4NyRGPkYsNyRGKEY6NyRGMkYxNyRGJUY3 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.2.12. Az Euler-f<Font encoding="UTF-8">\303\251le \317\206 f\303\274</Font>ggv<Font encoding="UTF-8">\303\251</Font>ny.</Text-field> phi(1); phi(2); phi(3); phi(4); phi(5); phi(6); phi(7); phi(8); IiIi IiIi IiIj IiIj IiIl IiIj IiIn IiIl LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.2.13. Lemma.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.14. Euler-Fermat t\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.15. K\303\266vetkezm\303\251ny: Fermat-t\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.16. Line\303\241ris kongruencia megold\303\241sa.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.17. P\303\251lda.</Font></Text-field> x:='x'; msolve(172*x=6,62); SSJ4RzYi NiQ8Iy9JInhHNiIiIiU8Iy9GJSIjTg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->6.2.18. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.2.19. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.20. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.21. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.22. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.23. Line\303\241ris kongruenciarendszer megold\303\241sa.</Font></Text-field> y:='y'; msolve({3*x-4*y=1,7*x+y=2},19); SSJ5RzYi PCQvSSJ5RzYiIiM2L0kieEdGJSIjOg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->6.2.24. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.25. Diofantikus probl\303\251m\303\241k.</Font></Text-field> isolve(x^2+y^2=z^2); PCUvSSJ4RzYiKihJJF9aM0dGJSIiIiwmKiQpSSRfWjFHRiUiIiNGKCEiIiokKUkkX1oyR0YlRi1GKEYoRigtSSVpZ2NkRyUqcHJvdGVjdGVkRzYlRiksJkYqRihGL0YoLCQqKEYtRihGLEYoRjFGKEYuRi4vSSJ6R0YlKihGJ0YoRjZGKEYyRi4vSSJ5R0YlLCQqLEYtRihGJ0YoRixGKEYxRihGMkYuRi4= isolve(x^3+y^3=z^3); LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.2.26. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.2.27. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.28. K\303\255nai marad\303\251kt\303\251tel.</Font></Text-field> chrem([1,2,2],[2,3,7]); IiNC LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.29. Megjegyz\303\251s.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.30. P\303\251lda.</Font></Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.31. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.2.32. Feladat.</Text-field> 42&^600 mod 13; IiIi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->6.2.33. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.2.34. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.2.35. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.36. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.37. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.38. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.2.39. Az RSA elj\303\241r\303\241s.</Font></Text-field> p:=safeprime(1563456788814256178886765661555261342987645321345665432123456788772091275121098761232122333321233434123432123344454321234320948725467845467788859812342365); log[2.](p); q:=safeprime(29841524475159001676561453467890987651234254321456490998767626788182514325678909987236514234154232396587874778993377722004988376667882767156363888377626677728888); log[2.](q); n:=p*q; e:=2876354132453678909987653432123409887635423125; igcdex(e,(p-1)*(q-1),'d'); d; d*e mod (p-1)*(q-1); ImV0ekpPNylmKSl5bmEleVlEKFs0S003S2FXTUJAVkJUVkw3S0xCN0s3dyk0QF5GIjRzKCl5Y003S2FtWDhLWHcpSE1oX2JoY3cnKSl5aEQ5KSl5Y01jIg== JCIrenQqKikzJiEiKA== Ilx1Rk4heW5Fd1ApKVFPY3J3Iyl5bXckKSlcK0F4UCQqKnlaKHllJ1JLVTpNVV5PcykqNCp5Y0s5RD0peUV3dykqNFxjOUthVUJedyk0KnlZYDljdzsrZl5aQzolKUg= JCIrayJlM0wmISIo ImVebExCOEpcUGg+W1AtRnBSbS5rPUVAZUkieW4ocGZNJVFiKEdqQEVjZDFyO0FHcmViNWxbcHltbm9FUExoSjR4IjMjcCNvTlYxXmpHKDNdKnlWIUcleW0hKTM6RDoqPVA7Njkob15KKVswbW9BJVtMYVwuJWZ1Smw2QS0nUmQ0ITQ1V1VQZ3VFaVgqPjVOYCVcTSJ5S3RGKkc2cy9Td1tBIz1DQ1RESFMkZmxZ Ik9ESlVOdykpNE03S01sKCkqNCp5T1hLVE53Rw== IiIi IWVebEpRWytVOSpvSzIzZyYpekp2P0Z0O0RsLzc1IW9WUHJvOlsncE8tVWUqMyk0Q0s9ZGxIOjNYXEJNOnlpLStrOFNeJypbPiNmJyo+cipIW1UqZmRNS3NCMmVyJCp5JlEnb05GQlQzS1xAI29TL2tNVXAneTw7KFwxUydIQjolZWEkRzEjPU9eN24/YiV5JykpPmtANihcSipSI3ooKltMOl1zRGQ2bVAnR3lDaUhyJj4= IiIi M:="Mint v\303\255z alatti, elmer\303\274lt harangok hint\303\241znak-e hajnalonk\303\251nt \303\241gyadn\303\241l a tizennyolc \303\251ves iskol\303\241sok kiket felakasztatt\303\241l"; convert(M,'bytes'); m:=sum(%[i]*256^(i-1),i=1..nops(%)); c:=m&^e mod n; UVx0TWludH5gdnxed3xodXpgfmFsYXR0aSx+YGVsbWVyfF53fGd2bHRgfmhhcmFuZ29rfCt8K2BoaW50fF53fFx1em5ha2AtZX5gaGFqbmFsb25rfF53fGR1bnRgfmB8Xnd8XHVneWFkbnxed3xcdWxgfCt8K2F+dGl6ZW5ueW9sY35gfF53fGR1dmVzYH5gaXNrb2x8Xnd8XHVzb2tgfCt8K2tpa2V0fmBmZWxha2FzenRhdHR8Xnd8XHVsYDYi N11zIiN4IiQwIiIkNSIiJDsiIiNLIiQ9IiIkJj4iJHQiIiRBIkYnIiMoKiIkMyJGLEYmRiZGJCIjV0YnIiQsIkYtIiQ0IkYvIiQ5IkYpIiQpPUYtRiZGJyIkLyJGLEYxRixGJSIkLiIiJDYiIiQyIiIjNUY3RjNGJEYlRiZGKSIkaCJGK0YlRixGNiIjWEYvRidGM0YsIiQxIkYlRixGLUY1RiVGNkYpIiRwIkYlRiZGJ0YpRjhGNCIkQCJGLCIkKyJGJUYpRjhGLUY3RjdGLEYnRiZGJEYrRi9GJUYlRjxGNUYtIiMqKkYnRilGO0YoRi8iJDoiRidGJEY/RjZGNUYtRilGOEY/RjVGNkY3RjdGNkYkRjZGL0YmRiciJC0iRi9GLUYsRjZGLEY/RitGJkYsRiZGJkYpRjhGLQ== ImJebCQqKT1jbCJ6JilvaDdJbCp6cUpuIilcYFtSeFZ4W3BmRmAoNHlcJipbIWZ3Kls1NClvREo6IXB4I3lyLUR6JG9AW2IzPEpTKlEiKj5BdTVHNm4qZiUpPl1ETkRKUSJmSVdmNFkoPmlvYSpHRU90XTUzRylIeWEiW3glUkpjLGFxYE4zQ3lWdWghcDZBdDdFQjZAVFkleSpRI2VNI1JnNCo9NnVTMDYxQ1krb0cmPg== ImNebD1LYlAoKilbQ1Y6JSlbJ2VFSyJRS0xbVDwiSDxrRlYpZiE+NzAmMyVIJ3lmOikzd2h3MWtQSjI+aTZKWk1cMS0vJylmZExYTihvR0hJemp6Z3VfNT8qcENaKDRVKG9pQXBGWlohelwxUlklW0AiPWtyMU0rJ0dxXi9BMEldIT0oKUhcWTEmRzcmXG1zOlFvdC9fZCNcZSkqKTRNUTxKISkpb15vdllFKXBBJVxbX3g= c&^d mod n; convert(%,base,256); convert(%,'bytes'); ImJebCQqKT1jbCJ6JilvaDdJbCp6cUpuIilcYFtSeFZ4W3BmRmAoNHlcJipbIWZ3Kls1NClvREo6IXB4I3lyLUR6JG9AW2IzPEpTKlEiKj5BdTVHNm4qZiUpPl1ETkRKUSJmSVdmNFkoPmlvYSpHRU90XTUzRylIeWEiW3glUkpjLGFxYE4zQ3lWdWghcDZBdDdFQjZAVFkleSpRI2VNI1JnNCo9NnVTMDYxQ1krb0cmPg== N11zIiN4IiQwIiIkNSIiJDsiIiNLIiQ9IiIkJj4iJHQiIiRBIkYnIiMoKiIkMyJGLEYmRiZGJCIjV0YnIiQsIkYtIiQ0IkYvIiQ5IkYpIiQpPUYtRiZGJyIkLyJGLEYxRixGJSIkLiIiJDYiIiQyIiIjNUY3RjNGJEYlRiZGKSIkaCJGK0YlRixGNiIjWEYvRidGM0YsIiQxIkYlRixGLUY1RiVGNkYpIiRwIkYlRiZGJ0YpRjhGNCIkQCJGLCIkKyJGJUYpRjhGLUY3RjdGLEYnRiZGJEYrRi9GJUYlRjxGNUYtIiMqKkYnRilGO0YoRi8iJDoiRidGJEY/RjZGNUYtRilGOEY/RjVGNkY3RjdGNkYkRjZGL0YmRiciJC0iRi9GLUYsRjZGLEY/RitGJkYsRiZGJkYpRjhGLQ== UVx0TWludH5gdnxed3xodXpgfmFsYXR0aSx+YGVsbWVyfF53fGd2bHRgfmhhcmFuZ29rfCt8K2BoaW50fF53fFx1em5ha2AtZX5gaGFqbmFsb25rfF53fGR1bnRgfmB8Xnd8XHVneWFkbnxed3xcdWxgfCt8K2F+dGl6ZW5ueW9sY35gfF53fGR1dmVzYH5gaXNrb2x8Xnd8XHVzb2tgfCt8K2tpa2V0fmBmZWxha2FzenRhdHR8Xnd8XHVsYDYi LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.2.40. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.2.41. A Miller-Rabin-f\303\251le val\303\263sz\303\255n\305\261s\303\251gi teszt.</Font></Text-field> millerrabin:=proc(n::posint,a::posint) local j,k,q,b; if n=2 or n=3 or n=5 or n=7 then return true fi; if n<9 then return false fi; b:=a mod n; if b=0 or b=1 then return FAIL fi; k:=0; q:=n-1; while type(q,even) do k:=k+1; q:=q/2; od; b:=b&^q mod n; j:=0; if b=1 then return true fi; while j<k do if b=n-1 then return true fi; b:=b^2 mod n; j:=j+1; od; false; end; Zio2JCdJIm5HNiJJJ3Bvc2ludEclKnByb3RlY3RlZEcnSSJhR0YmRic2JkkiakdGJkkia0dGJkkicUdGJkkiYkdGJkYmRiZDLkAkNTU1L0YlIiIjL0YlIiIkL0YlIiImL0YlIiIoT0kldHJ1ZUdGKEAkMkYlIiIqT0kmZmFsc2VHRig+Ri8tSSRtb2RHRiY2JEYqRiVAJDUvRi8iIiEvRi8iIiJPSSVGQUlMR0YoPkYtRks+Ri4sJkYlRk1GTSEiIj8oRiZGTUZNRiYtSSV0eXBlR0YoNiRGLkklZXZlbkdGKEMkPkYtLCZGLUZNRk1GTT5GLiwkKiYjRk1GNkZNRi5GTUZNPkYvLUZGNiQtSSMmXkdGJjYkRi9GLkYlPkYsRktAJEZMRj0/KEYmRk1GTUYmMkYsRi1DJUAkL0YvRlJGPT5GLy1GRjYkKiQpRi9GNkZNRiU+RiwsJkYsRk1GTUZNRkNGJkYmRiY= millerrabin(9,2); millerrabin(11,2); SSZmYWxzZUclKnByb3RlY3RlZEc= SSV0cnVlRyUqcHJvdGVjdGVkRw== for n do if millerrabin(n,2)<>isprime(n) then print(n) fi; od; IiVaPw== IiV4Sw== IiVMUw== IiUibyU= IiVAJCk= IiZUZSI= IiZUJEg= IiYqelU= IiZUIlw= IiZMRSY= IiYiR2w= IiZsWSg= IiYiZSEp IiYqWyYp IiZkJCkp IiZeMio= Warning, computation interrupted for n do if millerrabin(n,2)<>isprime(n) and millerrabin(n,3)<> isprime(n) then print(n) fi; od; IihgT1Ai IigoeUk6 IihAcSk+ IihgV0cj IigyaDYk Warning, computation interrupted n;
<Text-field style="Heading 3" layout="Heading 3">*6.2.42. Digital Signature Standard.</Text-field> StringTools[Hash](M); h:=convert(%,decimal,hex); UUE2ODQ4YjY1ZWU0MDhiN2VjY2I3NTIxYTdmMjU4ODg5NTYi IkhQJVtaMzNFQyg0QjotOT8mZUQ8J1Ei q:=nextprime(convert("97654376ad4efcbe43598123daf56c7b386acbda",decimal,hex)); for i from convert("ffffffffeeeeeeeeddddddddccccccccbbbbbbbbaaaaaaaa99999999888888887777777766666666555555554444444433333333222222221111111100000000ffffffffeeeeeeeeddddddddccccccccbbbbbbbbaaaaaaaa9999999988888888777777776666666655555555",decimal,hex) do p:=i*q+1; if isprime(p) then break fi; od: p; log[2.](q); log[2.](p); a:=2; g:=a&^((p-1)/q) mod p; x:=43276509876576543211245656730909809123093875; y:=g&^x mod p; IlFQPitxJGU7bUJQKGZxTTRQYVlIbSZlZUprKQ== ImBebFAhKj1ASEFHLj5DMUYhZnFNVHg7RVpfYjBZbTspPihldTAmKUhicXZdTnhFRjYyTmxnKypmdi1lXmhWXUFlVSZ5VVYpSCNHJipvJXo0Z2deUHNAXFE7YTcoSGNmdHg2bz0jKlJnKXBtZUhCbk5aRCc+RlNjXU8qKkg1UDwuRGslcEtXJ0d6SzglcCs4THlYaS96YXo9cyR5KGVDbGwkPjNmYlYieU5KMSI= JCIrInpAQ2YiISIo JCIrekBDQjUhIic= IiIj Il9ebDldLTBdWTphNlpAaipwX0Y2aWAnZiFvcSRIJmUyN1d3OW4nKVspZSFwcSZHKjRsU18oXDlcUiN5M0pMaXNbRlpLJnAycWBhMk8pPV1nSClRTFlDOWx6QzR1WlRUVGoxRD4tJ29nKFspPWpDazYjZkdTNlwmPlApKUglUnQlZjdOcG8+cFFWeWJlPyx0PjlYYjU4NCYzaSc0Ml9hSGgyTC90bCkpPit2ZCg= Ik12UTRCIjQpNDR0Y2NDNkthd2woKTRsRlY= Il9ebCJSZjAoUUJdLXFLLypRUlFiJmUoKVtsWiRlcGB1My45PiR6LXk9PV5kKDMlUiwqZWhcP0lxdnAkW2RrUytvKnBUQ0JyWClSLCUpZV1bPXNtOV4lUXBUKDQjUiRSS2YkSGktTTRpLj9qcz1TdDleJyk9RU8xVCNbYyYzNSVbTCI0SWhyXSQ+InAyW2p4VWY5dm8+T24jeiJSSi5pZHc3KSkqKlt6TzxQJykpM3RL k:=9804563211276543278906543278; r:=(g&^k mod p) mod q; s:=(h+x*r)/k mod q; Ij15S2ExKnlLYXc3QGpYISkq IlBgQFxkeXZccGBvLlxTbDlyZC5vXl5GQiQ= IlEiKVFdJlFxVjUkKjNMaWdpXTljLU9vNSNHQlk= w:=1/s mod q; u1:=h*w mod q; u2:=r*w mod q; v:=(g&^u1*y&^u2 mod p) mod q; IlFoNTdNem1qaF80OSg9KSpwTGdHc0hJOyMqKWU= IlFZI3koUSgpNFMiUSUqKilmY1wrKnlwKzMhKnk0UD4= IlFcJnB4ZF4qPlEkPWphbSY9NSdwbG8rRSE+Vzk= IlBgQFxkeXZccGBvLlxTbDlyZC5vXl5GQiQ=
<Text-field style="Heading 3" layout="Heading 3">6.2.43. Feladat: <Font encoding="UTF-8">\303\241</Font>lpr<Font encoding="UTF-8">\303\255</Font>mek.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.44. Feladat: Carmichel-sz<Font encoding="UTF-8">\303\241</Font>mok.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.2.45. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.46. Feladat: pitagoraszi sz<Font encoding="UTF-8">\303\241</Font>mh<Font encoding="UTF-8">\303\241</Font>rmasok.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.2.47. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.48. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.49. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.50. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.51. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok megold<Font encoding="UTF-8">\303\241</Font>sokkal.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.2.52. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok.</Text-field>
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">6.3. Sz\303\241melm\303\251leti f\303\274ggv\303\251nyek</Font></Text-field> restart: with(numtheory); 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
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.3.1. Sz\303\241melm\303\251leti f\303\274ggv\303\251nyek.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.3.2. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.3.3. P\303\251ld\303\241k.</Font></Text-field> divisors(60); tau(60); sigma[0](60); sigma(60); sigma[1](60); sigma[2](60); PC4iIiIiIiMiIiQiIiUiIiYiIiciIzUiIzciIzoiIz8iI0kiI2c= IiM3 IiM3 IiRvIg== IiRvIg== IiVnYQ== mobius(1); mobius(2); mobius(4); mobius(20); mobius(21); IiIi ISIi IiIh IiIh IiIi kappa:=proc(n::posint) nops(ifactors(n)[2]) end; kappa(1); kappa(2); kappa(3); kappa(4); kappa(5); kappa(6); Zio2IydJIm5HNiJJJ3Bvc2ludEclKnByb3RlY3RlZEdGJkYmRiYtSSVub3BzR0YoNiMmLUkpaWZhY3RvcnNHNiRGKEkoX3N5c2xpYkdGJjYjRiU2IyIiI0YmRiZGJg== IiIh IiIi IiIi IiIi IiIi IiIj nu:=proc(n::posint) nops(factorset(n)) end; factorset(60); nu(60); Zio2IydJIm5HNiJJJ3Bvc2ludEclKnByb3RlY3RlZEdGJkYmRiYtSSVub3BzR0YoNiMtX0kqbnVtdGhlb3J5RzYkRihJKF9zeXNsaWJHRiZJKmZhY3RvcnNldEdGJjYjRiVGJkYmRiY= PCUiIiMiIiQiIiY= IiIk LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.3.4. T\303\251tel.</Font></Text-field> phi(60); invphi(16); IiM7 NygiIzwiI0siI00iI1MiI1siI2c= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">6.3.5. P\303\251lda.</Font></Text-field> phi(5500); IiUrPw== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">->6.3.6. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.3.7. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.3.8. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.3.9. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.3.10. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.3.11. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.3.12. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.3.13. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.3.14. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">->6.3.15. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.3.16. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.3.17. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.3.18. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.3.19. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.3.20. Konvol\303\272ci\303\263.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.3.21. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.3.22. \303\226sszegz\303\251si f\303\274ggv\303\251ny.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.3.23 M\303\266bius-f\303\251le inverzi\303\263s formula.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.3.24. P\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.3.25. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.3.26. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.3.27. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.3.28. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.3.29. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok megold<Font encoding="UTF-8">\303\241</Font>sokkal.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.3.30. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok.</Text-field>
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2"><Font encoding="UTF-8">6.4. L\303\241nct\303\266rtek</Font></Text-field> restart; with(numtheory); 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
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.4.1. L\303\241nct\303\266rtek.</Font></Text-field> nextcfrac:=proc(L::list) local a,q,j; j:=nops(L); if j=0 then return FAIL fi; a:=L[j]; if type(a,integer) then return(L) fi; q:=floor(a); a:=a-q; a:=simplify(expand(1/a)); [op(L[1..j-1]),q,a]; end; [19/7]; nextcfrac(%); nextcfrac(%); nextcfrac(%); nextcfrac(%); cfrac(19/7); cfrac(19/7,quotients); Zio2IydJIkxHNiJJJWxpc3RHJSpwcm90ZWN0ZWRHNiVJImFHRiZJInFHRiZJImpHRiZGJkYmQyo+RiwtSSVub3BzR0YoNiNGJUAkL0YsIiIhT0klRkFJTEdGKD5GKiZGJTYjRixAJC1JJXR5cGVHRig2JEYqSShpbnRlZ2VyR0YoT0YlPkYrLUkmZmxvb3JHRiY2I0YqPkYqLCZGKiIiIkYrISIiPkYqLUkpc2ltcGxpZnlHRiY2Iy1JJ2V4cGFuZEdGKDYjKiRGKkZHNyUtSSNvcEdGKDYjJkYlNiM7RkYsJkYsRkZGRkZHRitGKkYmRiZGJg== NyMjIiM+IiIo NyQiIiMjIiIoIiIm NyUiIiMiIiIjIiImRiM= NyYiIiMiIiJGI0Yj NyYiIiMiIiJGI0Yj LUkmQ0ZSQUNHNiI2IzcmIiIjIiIiRidGJw== NyYiIiMiIiJGI0Yj LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.4.2. P\303\251ld\303\241k.</Font></Text-field> [172/62]; nextcfrac(%); nextcfrac(%); nextcfrac(%); nextcfrac(%); cfrac(172/62); NyMjIiMnKSIjSg== NyQiIiMjIiNKIiND NyUiIiMiIiIjIiNDIiIo NyYiIiMiIiIiIiQjIiIoRiU= NyciIiMiIiIiIiRGI0Yl LUkmQ0ZSQUNHNiI2IzcnIiIjIiIiIiIkRidGKQ== a:=(sqrt(5)+1)/2; nextcfrac([a]); nextcfrac(%); nextcfrac(%); cfrac(a); cfrac(a,periodic); cfrac(a,periodic,quotients); LCYqJiMiIiIiIiNGJSkiIiZGJEYlRiVGJEYl NyQiIiIsJComIiIjRiMsJiokKSIiJiNGI0YmRiNGI0YjISIiRixGIw== NyUiIiJGIywkKiYsJiokKSIiJiNGIyIiI0YjRiNGIyEiIkYjLCYiIiRGLEYnRiNGLEYs NyYiIiJGI0YjLCQqKCNGIyIiI0YjLCYiIiQhIiIqJCkiIiZGJkYjRiNGIywmRitGI0YnRipGKkYq LUkmQ0ZSQUNHNiI2IzcuIiIiRidGJ0YnRidGJ0YnRidGJ0YnRidJJC4uLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJA== LUkmQ0ZSQUNHNiI2IzckNyI3IyIiIg== NyQ3IjcjIiIi cfrac(sqrt(31),periodic); invcfrac(%); cfrac(3/5+sqrt(29),periodic,quotients); invcfrac(%); LUkmQ0ZSQUNHNiI2IzckNyMiIiY3KiIiIkYqIiIkRihGK0YqRioiIzU= KiQpIiNKIyIiIiIiI0Ym NyQ3IyIiJjdOIiIiIiNtIiIjRihGJCIjNUYmRiZGKEYoIiIkRihGJkYmRiYiJG8jRiZGJkYmRihGKkYoRihGJkYmRilGJEYoRihGJ0YmIiIqRiZGKkYmRiZGJiIiKUYmRiZGJkYqRiZGLA== LCYjIiIkIiImIiIiKiQpIiNII0YmIiIjRiZGJg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">*6.4.3. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.4. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.5. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.6. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.4.7. L\303\241nct\303\266rtk\303\266</Font>zel<Font encoding="UTF-8">\303\255t\303\251sek z\303\241rt alakja.</Font></Text-field> cfrac(3/5+sqrt(29)); nthconver(%,7); nthnumer(%%,7); nthdenom(%%%,7); LUkmQ0ZSQUNHNiI2IzcuIiImIiIiIiNtIiIjRipGJyIjNUYoRihGKkYqSSQuLi5HNiQlKnByb3RlY3RlZEdJKF9zeXNsaWJHRiQ= IyInUz03IiZkLiM= IidTPTc= IiZkLiM= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.4.8. P\303\251lda.</Font></Text-field> cfrac(172/62); nthconver(%,4); nthnumer(%%,4); nthdenom(%%%,4); LUkmQ0ZSQUNHNiI2IzcnIiIjIiIiIiIkRidGKQ== IyIjJykiI0o= IiMnKQ== IiNK LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.4.9. Megjegyz\303\251s.</Font></Text-field> for n from 0 do expand(1/sqrt(5)*(((1+sqrt(5))/2)^n-((1-sqrt(5))/2)^n)) od; IiIh IiIi IiIi IiIj IiIk IiIm IiIp IiM4 IiNA IiNN IiNi IiMqKQ== IiRXIg== IiRMIw== IiR4JA== IiQ1Jw== IiQoKSo= IiUoZiI= IiUlZSM= IiUiPSU= IiVsbg== IiZZNCI= IiY2eCI= IiZkJ0c= IiZvaiU= IiZEXSg= IickUkAi Iic9az4= Iic2eUo= IidIVV4= IidTPyQp IihwaU0i Iig0JHlA Iih5WF8k IigoKUdxJg== IihsdUEq IilfLiRcIg== Iik8eTpD IilwIikzUg== IiknKWZDag== IipiVEwtIg== IipULGVsIg== IionSDl6RQ== IipQV1xMJQ== IipMKDM5cQ== IitxSiFcOCI= IisuPkpPPQ== Iit0XUBySA== Iit3cF8yWw== IitcP3V5eA== Warning, computation interrupted LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.4.10. Megjegyz\303\251s.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.4.11. T\303\251tel.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3"><Font encoding="UTF-8">*6.4.12. P\303\251lda.</Font></Text-field> nextrangecfrac:=proc(L::list) local a,a1,a2,j,q1,q2; j:=nops(L); if j=0 then return FAIL fi; a:=L[j]; a1:=op(1,a); a2:=op(2,a); q1:=floor(a1); q2:=floor(a2); if q1<>q2 then return L fi; [op(L[1..j-1]),q1,1/(a2-q2)..1/(a1-q1)] end; [314159265/10^8..314159266/10^8]; nextrangecfrac(%); nextrangecfrac(%); nextrangecfrac(%); nextrangecfrac(%); nextrangecfrac(%); nextrangecfrac(%); evalf(355/113); Zio2IydJIkxHNiJJJWxpc3RHJSpwcm90ZWN0ZWRHNihJImFHRiZJI2ExR0YmSSNhMkdGJkkiakdGJkkjcTFHRiZJI3EyR0YmRiZGJkMrPkYtLUklbm9wc0dGKDYjRiVAJC9GLSIiIU9JJUZBSUxHRig+RiomRiU2I0YtPkYrLUkjb3BHRig2JCIiIkYqPkYsLUY/NiQiIiNGKj5GLi1JJmZsb29yRzYkRihJKF9zeXNsaWJHRiY2I0YrPkYvLUZINiNGLEAkMEYuRi9PRiU3JS1GPzYjJkYlNiM7RkEsJkYtRkFGQSEiIkYuOyokLCZGLEZBRi9GWUZZKiQsJkYrRkFGLkZZRllGJkYmRiY= NyM7IyIpYD0kRyciKSsrKz8jIipMJ3pxOiIpKysrXQ== NyQiIiQ7IyIpKysrXSIoTCd6cSMiKSsrKz8iKGA9JEc= NyUiIiQiIig7IyIoYD0kRyInSHE8IyIoTCd6cSIncERX NyYiIiQiIigiIzo7IyIncERXIicpNFQlIyInSHE8Iic9azw= NyciIiQiIigiIzoiIiI7IyInPWs8IiQ2JyMiJyk0VCUiJXI5 NyciIiQiIigiIzoiIiI7IyInPWs8IiQ2JyMiJyk0VCUiJXI5 NyciIiQiIigiIzoiIiI7IyInPWs8IiQ2JyMiJyk0VCUiJXI5 JCIrP0hmVEohIio= cfrac(Pi,100,quotients); N2JxIiIkIiIoIiM6IiIiIiQjSEYmRiZGJiIiI0YmRiNGJiIjOUYoRiZGJkYoRihGKEYoRiYiIyUpRihGJkYmRiVGIyIjOEYmIiIlRigiIidGLSIjKipGJkYoRihGLUYjIiImRiZGJkYtIiIpRiZGJEYmRihGI0YkRiZGKEYmRiYiIzdGJkYmRiZGI0YmRiZGMEYmRiZGKEYmRi1GJkYmRi9GKEYoRiNGJkYoRixGLCIjO0YmIiRoIiIjWEYmIiNBRiZGKEYoRiZGLEYmRigiI0NGJkYoRiZGI0YmRihGJkYmIiM1RihJJC4uLkc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkc2Ig== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">6.4.13. Intervallumaritmetika.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.14. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.15. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.16. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.17. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.18. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.19. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.20. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.21. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.22. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.4.23. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.4.24. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.4.25. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.4.26. Feladat.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">6.4.27. Feladat.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 3" layout="Heading 3">*6.4.28. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok megold<Font encoding="UTF-8">\303\241</Font>sokkal.</Text-field>
<Text-field style="Heading 3" layout="Heading 3">*6.4.29. Tov<Font encoding="UTF-8">\303\241</Font>bbi feladatok.</Text-field>
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">7. Gr\303\241felm\303\251let</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">8. Algebra</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">9. K\303\263dol\303\241s</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">10. Algoritmusok</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn