Prove the following statements using direct, contrapositive, or contradiction proof: a. If n is an integer, then 4 | n^2 or 4 | [(n^2) + 1]. b. The product of any n consecutive positive integers is divisible by n!. c. Suppose a,b, and p are integers and p is prime. Prove that if p | ab then p | a or p | b. d. If n is an integer, then gcd(n,n + 2) = {1,2}. e. Suppose a, and b are integers. Then a = lcm(a,b) if and only if b | a.

Prove the following statements using direct, contrapositive, or contradiction proof: a. If n is an integer, then 4 | n^2 or 4 | [(n^2) + 1]. b. The product of any n consecutive positive integers is divisible by n!. c. Suppose a,b, and p are integers and p is prime. Prove that if p | ab then p | a or p | b. d. If n is an integer, then gcd(n,n + 2) = {1,2}. e. Suppose a, and b are integers. Then a = lcm(a,b) if and only if b | a.

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter2: The Integers

Section2.4: Prime Factors And Greatest Common Divisor

Problem 28E: Let and be positive integers. If and is the least common multiple of and , prove that . Note...

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