1.6 Divisibility in the Integers

Definition 1.6.1. Let a and b be integers. We say that a divides b, written a | b,
if there exists an integer k such that b = a k.

Theorem 1.6.2. Let a, b and c be integers. If a divides b and a divides c, then a
divides b + c.

Proof. Write b = a k and c = a m for integers k and m. Then b + c = a (k + m), so a
divides b + c.

Theorem 1.6.3. Let a and b be integers, not both zero. Then the set of common
divisors of a and b has a greatest element, written gcd(a, b), and there exist
integers s and t with gcd(a, b) = s a + t b.

Exercise 1.6.4. Show that if a divides b and b divides a, then a = b or a = -b.

Exercise 1.6.5. Prove that the sum of two odd integers is even.

Exercise 1.6.6. Determine whether every integer greater than 2 is the sum of two
primes.