how can I prove that |Um*n| = |Um| cross |Un| implies gcd(m,n) =1 ?

where |Um*n| denotes all the numbers less than m*n that are relatively prime to it.

my lecturer proved us the other implication , which is gcd(m,n)=1 implies |Um*n| = |Um| cross |Un| using the chinese remainder theorem but said there is an equivalence between the two.

I tried to prove it the other direction myself , but it seems I have something missing.

Thank you for helping me .