1. ## 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.

2. Let $d|(m\cdot{n})$ and $d=d_1\cdot{d_2}=d_3\cdot{d_4}$ where $d_1|m,d_3|m$ and $d_2|n,d_4|n$

we have: $
d_1 \cdot d_2 = \dot d_3
$
and since $(d_2,d_3)=1$ ( because $(n,m)=1$ ) it follows that $d_1=\dot d_3$ (1)

Again $
d_3 \cdot d_4 = \dot d_1
$
and since $(d_4,d_1)=1$ it follows that $d_3=\dot d_1$ (2)

By (1) and (2) we must have (think about the quotient) $d_1=d_3$ and from there $d_2=d_4$ which proves that the representation is unique