# HW 3 - MATH 290 HOMEWORK 3 SOLUTIONS 1 Prove that p is...

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MATH 290 – HOMEWORK 3 – SOLUTIONS 1. Prove that p is prime if and only if for all natural numbers a and b , either p divides a or p divides b whenever p divides ab . (Hint: You may use the deﬁnition of a prime number, the facts given in Exercises 9(a) and 9(b) of Section 1.7, page 65, and after trying hard to solve the problem yourself , you may consult the hint given for Exercise 9(c) in the back of the book. You may not use the fact that every natural number can be written as a product of primes.) Solution. (= ) Suppose that p is prime. We must show that for all natural numbers a and b , if p divides ab then either p divides a or p divides b . So suppose that natural numbers a and b are given, that p divides ab and that p does not divide a . We must show that p divides b . By Exercise 9(b), since p does not divide a , GCD ( p,a ) = 1. This means that there are integers x and y such that px + ay = 1. Multiplying both sides by

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HW 3 - MATH 290 HOMEWORK 3 SOLUTIONS 1 Prove that p is...

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