Friday, June 1, 2018

elementary number theory - Prove if nmidab, then nmid[gcd(a,n)timesgcd(b,n)]

Prove if nab, then n[gcd(a,n)×gcd(b,n)]



So I started by letting d=gcd(a,n) and e=gcd(b,n).
Then we have x,y,w,z so that dx=a, ey=b,dw=ez=n
and we also have s so that ns=ab



or ns=dexy.



what I want is nde, but I'm only getting to nde(xy) since I cannot prove that s/(xy) is an integer.

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