Use mathematical induction to prove the second part of Corollary 8.15. Let p be

Question

Use mathematical induction to prove the second part of Corollary 8.15. Let p be a prime number, let n N. and let a1, a2, ..., an Z. If P | (a1a2 , ... ,an), then there exists a k N with 1 le k le n such that P | ak. Let a and b be nonzero integers. If there exist integers x and y such that ax + by = 1. what conclusion can be made about gcd (a, b)? Explain. Let a and b be nonzero integers. If there exist integers x and y such that ax + by = 2, what conclusion can be made about gcd ((a, b)? Explain. Let a Z. What is gcd (a, a + 1)? That is, what is the greatest common divisor of two consecutive integers? Justify your conclusion.

Explanation / Answer

(Base case) If n = 1, then p | a_1 so p | a_1. (Inductive step) Assume for some positive integer k, if p | (a_1 * a_2 * ... * a_k), then for some 1

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