![]() ![]() Greatest Common Divisor gcdgcd(343, 63)=7, gcd(12345,11111)=1 gcd(1993,3980021)=1993Euclidean Algorithm to compute gcd(a,b) does not require the factorization of the numbers and is fast.gcd(482,1180)=2 This factorization into primes is unique, up to reordering the factors49500=22 32 5311If a prime p|ab, then either p|a or p|b Moreover, p|x1 x2 xn p|xj for some j7|1430, Prime Factorization TheoremEvery positive integer is a product of primes. Prime NumbersAn integer p>1 that is divisible only by 1 and itself is called a prime number, otherwise it is called composite (P.64)primegen.c generates prime numbersLet (x) be the number of primes less than x, then (x) x/ln(x) as åxercise Plot (x) vs. ![]() if there exists an integer k such that b=ka, we say a divides b which is denoted by a|b 11|143, 1993|3980021 if a0, then a|0 and a|a 1|b for each b a|b and b|c a|c a|b and a|c a|sb+tc for all s, t Basic Number TheoryDivisibility Let a,b be integers with a0. ![]()
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