By Raphael Pass, Wei-lung Tseng

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The next theorem shows that addition and multiplication “carry over” to the modular world (specifically, addition and multiplication can be computed before or after computing the remainder). 17. If a ≡ b (mod m), and c ≡ d (mod m) then 1. a + c ≡ b + d (mod m) 2. ac ≡ bd (mod m) Proof. 16, we have unique integers r and r such that a = q1 m + r b = q2 m + r c = q1 m + r d = q2 m + r This shows that ac = q1 m · q1 m + q1 mr + q1 mr + rr bd = q2 m · q2 m + q2 mr + q2 mr + rr which clearly implies that m|(ac − bd).

2 Modular Arithmetic Modular arithmetic, as the name implies, is arithmetic on the remainders of integers, with respect to a fixed divisor. A central idea to modular arithmetic is congruences: two integers are considered “the same” if they have the same remainder with respect to the fixed divisor. 15. Let a, b ∈ Z, m ∈ N+ . , if there exists k ∈ Z such that a − b = km). As a direct consequence, we have a ≡ a (mod m) for any m ∈ N+ . 16. , a mod m = b mod m. Proof. We start with the if direction.

A central idea to modular arithmetic is congruences: two integers are considered “the same” if they have the same remainder with respect to the fixed divisor. 15. Let a, b ∈ Z, m ∈ N+ . , if there exists k ∈ Z such that a − b = km). As a direct consequence, we have a ≡ a (mod m) for any m ∈ N+ . 16. , a mod m = b mod m. Proof. We start with the if direction. Assume a and b have the same remainder when divided by m. That is, a = q1 m + r and b = q2 m + r. Then we have a − b = (q1 − q2 )m ⇒ m|(a − b) 42 number theory For the only if direction, we start by assuming m|(a−b).

### A Course in Discrete Structures by Raphael Pass, Wei-lung Tseng

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