three problems I am stuck on
1. Find all pairs of positive integers 0 < a < b such that gcd(a,b) = 6 and ab = 2160
obviously I could list all integers of this form and check each, I am wondering if there is a more theoretical, efficient way?
2. Let m and n be two integers such that the greatest common divisor, of m and n, gcd(m,n) = 1. What values can the greatest common divisor of (m + n), (m + 7n) take? i.e what values can gcd (m+n, m+7n) take? Why?
3. Let a,b,m,n be positive integers such that a|m (a divides m) and b|m and gcd (m/a, m/b) = 1. Prove that if a|n and b|n, then m|n