Integers, Prime Factorization, and More on Primes

The integer q is called the quotient and r is the remainder. Proof. Consider the rational number b a . Since R = ⋃ k∈Z[k, k + 1) (disjoint), there exists a unique integer q such that b a ∈ [q, q + 1), i.e., q ≤ b a < q + 1. Multiplying through by the positive integer a, we obtain qa ≤ b < (q + 1)a. Let r = b− qa. Then we have b = qa + r and 0 ≤ r < a, as required. Proposition 3. Let a, b, d ∈ Z. If d | a and d | b, then d | (ma + nb) for all m,n ∈ Z. Proof. Since d | a and d | b, there exist integers c1 and c2 such that a = c1d and b = c2a. Then for any integers m,n ∈ Z, we have ma + nb = mc1d + nc2d = (mc1 + nc2)d.