Thread: Number Theory (8)

Let m and n be positive integers with (m,n) =1. Prove that each divisor d>0 of mn can be written uniquely as d1d2 where d1,d2>0, d1 divides m, d2 divides n, and (d1,d2) =1 and each such product d1d2 corresponds to a divisor d of nm.

Let and where and

we have: and since ( because ) it follows that (1)

Again and since it follows that (2)

By (1) and (2) we must have (think about the quotient) and from there which proves that the representation is unique