Given $\gcd (a,b,c)=12$, find $\gcd (a,b ,c,16)$

Question

Given $\gcd (a,b,c)=12$, find $\gcd (a,b ,c,16)$ I need the rule or algorithm to solve similar questions. Is that $16 \bmod 12 = 4$ will be the new gcd of $(a,b ,c,16)$.

Answer 1

Yes, more precisely, since gcd is associative we have

$$\begin {align} \gcd(a,b,c,\,16) &= \gcd(\gcd(a,b,c),\,16)\\ &= \gcd(12,16)\\ &= \gcd(12,\,16\bmod 12))\ \ \rm by\ Euclid\\ &= \gcd(12,4) = 4\end{align}\qquad$$