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Congruences Robert Friedman Definition of congruence mod n Congruences are a very handy way to work with the information of divisibility and remainders, and their use permeates number theory. Definition 1. Let n be a positive integer (the modulus). We say that two integers a, b are congruent mod n, which is written as a ≡ b (mod n), if n|b − a. Example 2. 1. If a and b are arbitrary integers, a ≡ b (mod 1), since 1 divides every integer and in particular it divides b − a. 2. For n = 2, two integers a and b are congruent mod 2 if and only if their difference b − a is even. This happens exactly when a and b are both even or they are both odd. 3. Something similar happens for n = 3. Every integer has remainder 0, 1 or 2 when divided by 3, and it is easy to check that a ≡ b (mod 3) if and only if a and b have the same remainder when divided by 3. In fact, this generalizes: As we have seen, given integers n > 0 and a, there exist unique integers q, r with 0 ≤ r ≤ n − 1, such that a = nq + r. Here, r is the remainder when you divide a by n. With this said, we have the following alternate way to describe congruences: Proposition 3. Two integers a, b are congruent mod n if and only if they have the same remainder when divided by n. Proof. First suppose that a, b are congruent mod n. Thus, b − a = nk for some integer k, so that b = a + nk. Now long division with remainder says that a = nq + r, with 0 ≤ r ≤ n − 1. Hence, b = a + nk = nq + r + nk = nq + nk + r ...
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