I need help on a number theory question in where you Let a > 0, Show that (a,b) = a if and only if a | b?

Answer 1

Suppose (a, b) = a. Then by definition a divides b.

Now suppose a | b. Then clearly a | a and a | b, so a is a common divisor of a and b. But no number greater than a can divide a, so there can be no common divisor of a and b greater than a. So a = (a, b).