Let a,b,c be integers. If (b,c)=1, then (a,bc)=(a,b)(a,c)
This is difficult to answer because some theorems that we haven't proven yet, we can't use.
The Attempt at a Solution
Let g=(a,b) and h=(a,c), g and h are integers.
That means g|a and g|b, h|a and h|c.
a=gk, for some integer k
b=gq, for some integer q
a=hp for some integer p
c=ht for some integer t.
Multiply b and c together to get bc=(a,b)(a,c)qt. Then (a,bc)= (gk, (a,b)(a,c)qt), but that's as far as I got.