Divizibilitate în mulțimea numerelor naturale/întregi

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1 Divizibilitate în mulțimea numerelor naturale/întregi Teorema îmărţirii cu rest în mulțimea numerelor naturale Fie a, b, b 0. Atunci există q, r astfel încât a=bq+r, cu 0 r < b. În lus, q şi r sunt unic determinate de a şi b. Reciroc, dacă a, b, b 0 si q, r astfel încât a=bq+r, cu 0 r < b, atunci q,r sunt câtul resectiv restul îmartirii lui a la b. Teorema îmărţirii cu rest în mulțimea numerelor întregi Fie a, b, b 0. Atunci există q și r astfel încât a=bq+r, cu 0 r < b. În lus, q şi r sunt unic determinate de a şi b. Reciroc, dacă a, b, b 0 si q și r astfel încât a=bq+r, cu 0 r < b, atunci q,r sunt câtul resectiv restul îmartirii lui a la b. Definiţii Fie a, b. Sunem că a divide b dacă există c astfel încât b=ac ( a divide b se va nota ab sau b a) Sunem că numărul natural d este cel mai mare divizor comun al lui a și b și notăm d=(a,b) dacă: da şi db 2) dacă ca şi cb, atunci cd (c ) Curriculum for National Olymics Curriculum for International Olymics Page

2 Sunem că numărul natural m este cel mai mic multilu comun al lui a și b și notăm m=[a,b] dacă: am şi bm 2) dacă ac şi b c atunci mc (c ). Prorietăţi uzuale Fie a,b,c. aa (reflexivitatea) ab şi ba a = b ab şi bc ac (tranzitivitatea) 2) a, a a= ; a0, 0a a=0 3) ab şi ac a(xb±yc), () x, y 4) ab acbc ; acbc, c 0 ab 5) ac si bd abcd 6) ab abc ; ( Gauss) abc şi (a,b)= ac ab, rim a sau b 7) ac și bc, (a,b)= abc 8) (a,b)=, (a,c)= (a,bc)= ; (ca,cb)=c(a,b) 9) (a,b)=d () x, y astfel încât (x,y)= şi a=dx, b=dy;m=dxy Curriculum for National Olymics Curriculum for International Olymics Page 2

3 [a,b]=m () x, y astfel încât (x,y)= şi m=ax, m=by 0) [a,b](a,b) = ab dacă a,b,c și a=bc+r, r, atunci (a,b)=(b,r) (v. algoritmul lui Euclid de aflare a c.m.m.d.c. a două numere naturale) 2) (a,b)=d () x, y astfel încât d=ax+by (a,b)= () x, y astfel încât =ax+by 3) (a+b) n = M a+b n (a+b) n = M a+b n, n, n ar (a-b) n = M a- b n, n, n imar 4) ca-b c (a n b n ), n Criterii de divizibilitate Fie a. Atunci: a 2 u(a)ϵ0,2,4,6,8 a 4 numărul determinat de ultimele două cifre ale lui a este divizibil cu 4 (generalizare) a 5 u(a) 0,5 a 25 numărul determinat de ultimele două cifre ale lui a este divizibil cu 25 (generalizare) Curriculum for National Olymics Curriculum for International Olymics Page 3

4 a 0 u(a)=0 a 00 ultimele două cifre ale numărului a sunt 0 (generalizare) a 3 suma cifrelor lui a este divizibila cu 3 a 9 suma cifrelor lui a este divizibila cu 9 a diferența dintre suma cifrelor de rang ar și suma cifrelor de rang imar este divizibilă cu. Teorema fundamentală a aritmeticii Dacă n, n 2, atunci n oate fi descomus în mod unic (ână la o ermutare a factorilor) ca rodus finit de numere rime, i.e.: a a2 numere rime distincte, a a a a.î.: 2,,,...,,,..., 2 Numărul/ suma/ rodusul divizorilor unui număr natural... a n 2. a a2 a Dacă n, n 2 și este descomunerea în factori rimi a lui n, atunci: n... n 2 i.e. numărul divizorilor lui n este dat de: ( )... n a a2 a n i.e. suma divizorilor lui n este dată de: a a2 a 2 ( n)... 2 ni.e. rodusul divizorilor lui n este dat de: n 2 n a a2... a Teorema lui Legendre Exonentul unui număr rim din descomunerea în factori rimi a numărului n! este egal cu: n n n Congruențe în mulțimea numerelor naturale/întregi Curriculum for National Olymics Curriculum for International Olymics Page 4

5 Fie n. Dacă ab, sunem că a este conguent cu b modulo n dacă n a b. Deci mod n a b a b n sau altfel sus abdau, același rest la îmărțirea cu n. Prorietăți uzuale a amod n (reflexivitatea) 2) a bmod n b amod n (simetria) 3) a bmod n, b cmod n a cmod n (tranzitivitatea) 4) a bmod n, c d mod n a c b d mod n, ac bd mod n m m 5) a bmod n a b mod n 6) ac bcmod n, c, n a bmod n Sistem comlet de resturi modulo n Fie n. Mulțimea de numere întregi a, a2,..., an s.n. sistem comlet de resturi modulo n dacă ai nu este congruent cu a j modulo n entru i j. Evident mulțimea 0,,2,..., n constituie un sistem comlet de resturi modulo n și orice mulțime de n numere întregi consecutive constituie la rândul său un sistem comlet de rest modulo n. Teoremă Fie a, a2,..., an un sistem comlet de resturi modulo n și x, y, x, n. Atunci xa y, xa2 y,..., xan yeste un sistem comlet de resturi modulo n. Indicatorul lui Euler al unui număr natural Dacă n, vom nota cu n numărul de numere naturale mai mici ca n și rime cu n. Numărul n s.n. indicatorul lui Euler al numărului n. Curriculum for National Olymics Curriculum for International Olymics Page 5

6 Teoremă ( Prorietățile numărului,, a) b) ab a b a b a b c) n,,,, n... 2 n ) număr rim, a a2,... a n 2 fiind descomunerea în factori rimi a lui n. Teorema lui Euler Fie n și a astfel ca an,. Atunci a n mod n. Teorema lui Fermat Fie un număr rim și a astfel ca nu divide a. Atunci a mod Teorema lui Wilson Un număr natural este rim d.n.d.! mod Postulatul lui Bertrand.. Pentru orice n, n 2, între n și 2n se află cel uțin un număr rim. Teorema chinezească resturilor Fie n, a, b, i, n astfel ca numerele b i să fie rime între ele două câte două. i i Atunci sistemul de congruențe x a modb x a modb 2 2 x a modb are soluții. În lus, toate soluțiile sunt congruente între ele modulo b b 2... b n. n n Curriculum for National Olymics Curriculum for International Olymics Page 6

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