Use some algebraic factorization techniques to prove the following classic gems of number theory:
- Given two integers a and m larger than 1, show that, if m is odd, then is a divisor of . Use this result to obtain the factorization of 1001.
- Generalize, the above result to obtain that if a and m are two integers larger than 1 and if is an odd divisor of m, then is a divisor of . use this result to show that 101 is a factor of 1000001.
- Show that 7, 11, and 13 are factors of
- Show that is a composite number for each integer . More generally, show that if a is a positive integer such that is a perfect square, then is a composite number provided that .