Homework for Chapter 6: (To be collected on Tuesday, Sept. 13)

1. Use Theorem 1 to prove the following theorem:

Theorem. If a and b are integers, not both 0, then a positive integer d is the greatest common divisor of a and b if and only if:

(i) d| a and d| b.

(ii) if c is an integer with c | a and c | b, then c | d.

2. Prove that if p is a prime number and that p divides the product ab, then either p divides a or p divides b. (A prime number is a number p ≥ 2 whose only positive divisors are 1 and p. Numbers m ≥ 2 that are not primes are called composite numbers.)

NOTE: When writing up your proofs be sure to write out the theorem statement followed by a clear label of "Proof". Then proceed with your proof. Your presentation should be clear and concise, and your writing should be legible. You will be lose points for messy, convoluted proofs. If you want to use LaTeX to type up your work, that would be great. I can help you to get started on this if you have never used the program before. Stop by my office!

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