$a\mid b,c\mid d\Rightarrow\,\gcd(a,c)\mid \gcd(b,d)$

So far I have that $a|b$ implies $b=ax$ for some x in the integers and $c|d$ implies that $d=cy$ for some y in the integers. From here I can see that $gcd(a,c)|gcd(b,d)$ is logically equivalent to $gcd(a,c)|gcd(ax,cy)$ but I am not sure where to go from there.

If $a\mid b$ and $c\mid d$, then, since $\gcd(a,c)\mid a$ and $a\mid b$, $\gcd(a,c)\mid b$. For the same reason, $\gcd(a,c)\mid d$. So, since $\gcd(a,c)$ divides both $b$ and $d$, it divides $\gcd(b,d)$.

Sines and cosines of angles in arithmetic progression [closed]

Dot product of the gradient of a function

Prove that $x^2 + y^2 = 3(z^2 + m^2)$ has no solutions in integer [closed]

Does $a\mid bc$ imply $a\mid b$ or $a\mid c$?

$(a^{2^n}+1,a^{2^m}+1)=1 or 2$ [duplicate]

two functions are uniformly continuous on some interval I and each is bounded on I then their product is also uniformly continuous on I . [closed]

Infinite series $\sum_{k=1}^\infty \frac{k}{2^k}$ and $\sum_{k=1}^\infty \frac{k^2}{2^k}$ [duplicate]

Differentiation definition for spaces other than $\mathbb{R}^n$

Proving a function is constant, under certain conditions? [duplicate]

For which positive integer values of $n$ does $a|mn-b?$

f is differentiable on convex domain D and Re(f')>0 implies f is injective on D

$\alpha$ and $\beta$ are solution of $a \cdot \tan\theta + b \cdot \sec\theta = c$ show $ \tan(\alpha + \beta) = \frac{2ac}{a^2 - c^2}$