Find each of the following greatest common divisors by listing all of the common

Question

Find each of the following greatest common divisors by listing all of the common divisors of each pair of integers. gcd(21. 28) gcd(58.63) gcd(110.215) gcd(l 10,-215) Let a Z and let k Z with k 0. Prove that if k | a and k | (a + 1), then k | 1. and hence k = plusminus 1. Let a Z, Find the greatest common divisor of the consecutive integers a an.a + 1. That is. determine gcd(a, a + 1). Let a Z and let k Z with k 0. Prove that if k| a and k |(a + 2), then k | 2. Let a Z. What conclusions can be made about the greatest common divisor of a and a + 2? Let a, b Z with b 0. Prove each of the following:

Explanation / Answer

2B The gcd between two numbers is the largest integer that divides both numbers. So gcd(a,a+1) must be a number dividing a and a+1. But if the number divides a and a+1, then it must divide 1. But the only positive integer dividing 1 is itself. So: gcd(a,a+1) = 1

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