If we have integers $h$, $i$, $j$, and $k$, would it be true to say that $\gcd(h,i)\gcd(j,k)|\gcd(hi,jk)$? If so, how can we prove it?
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No, consider $h=i=2$ and $j=k=3$, then $$\gcd(h,i)\gcd(j,k)=\gcd(2,2)\gcd(3,3)=2\cdot 3=6\nmid 1=\gcd(4,9).$$
By distributivity $\rm\ (h,i)\:(j,k) = (hj,hk,ij,ik)\ $ which divides $\rm\ hj, ik\ $ so also $\rm\:(hj,ik)\:.\:$ Presumably that's what was intended.