**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**.)