Proving n & m Divide k When gcd(m,n)=1

[Diophantus]

I forgot how to prove the following, can anyone jog my memory:

Let m,n be natural numbers s.t. gcd(m,n) = 1 and n|k and m|k for some natural number k.

Then nm|k.

I know I'll kick myself when I find out. Thanks.

 

[neurocomp2003]

well what do n|k and m|k represent and what does it imply about k? think in terms of mathematical symbols and not words. Apply the gcd constriction and then you should be done.

 

[slider142]

This also falls to the fundamental theorem of arithmetic nicely.

 

[Diophantus]

Oh I see it now!

You just decompose m and n into their unique prime factors and note that no prime factor of m can be a prime factor of n and vice versa.

So if am = k = bn then each prime factor of n, say, divides a (with aprropriate multiplicity) and hence n divides a giving the result.

Ta.