By Raphael Pass, Wei-lung Tseng
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Extra resources for A Course in Discrete Structures
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