# Divisibility

May 8th, 2015
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• Definition: If n and d are integers and d ≠ 0, then n is divisible by d provided n = d ⋅ k for some integer k. • Alternatively, we say: n is a multiple of d d is a factor of n d is a divisor of n d divides n (denoted with d | n).

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Maths3.3.1Section 3 - Divisibility Definition: If n and d are integers and d 0, then n is divisible by d provided n = d k for some integer k. Alternatively, we say: n is a multiple of d d is a factor of n d is a divisor of n d divides n (denoted with d | n).3.3.2Properties of Divisibility Divisors of 0: If k is a non-zero integer, then k divides 0 since 0 = k 0. Positive Divisors of a Positive Number: If a and b are positive integers and a | b, is a b? Yes. Since a | b, k Z,such that b = a k. Moreover, 0 < k, since a and b are, so 1 k. Thus: a = a 1 a k = b. Therefore a b. Divisors of 1: The only divisors of 1 are 1 and 1.3.3.3Divisibility of Algebraic Terms Let a and b be integers. Does 3 | (3a + 3b)? Yes, since (3a + 3b) = 3(a + b) and (a + b) Z. Does 5 | 10ab? Yes again, since 10ab = 5(2ab) and (2ab) Z. If m Z and m | (a + b), does m | a and m | b? No. 2 | 8 but 2 | 5 and 2 | 3.3.3.4Divisibility and Non-divisibility There is another way to test for divisibility: If d | n, there is integer k with n = dk, then k = (n/d). So, if (n/d) is an integer, then d | n. This leads to an easy way to test for nondivisibility: If (n/d) is not an integer, then d cannot divide n. Examples: 3 | 12 since 12/3 = 4 Z. 5 | 12 since 12/5 = 2.4 Z.3.3.5Proving Properties of Divisibility Theorem: Transitivity of Divisibility For all a,b,c Z, if a

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