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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 $(a,b ,c,16)$.

Bernard
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geektobe
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    The fact that $16$ mod $12$ gives you the right answer is a coincidence. It would be the same answer with $16$ replaced by $20$. – Matt Samuel Oct 09 '19 at 17:41
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    Hint: $\gcd(a,b,c)=\gcd(\gcd(a,b),c)$, apply multiple times. –  Oct 09 '19 at 17:42
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    Welcome to MSE. You should choose your tags carefully. What has this to do with calculus or with education? – José Carlos Santos Oct 09 '19 at 17:44
  • you meen i will take the gcd( 12,16) = 4 ? – geektobe Oct 09 '19 at 17:45
  • @José Carlos Santos : ops its my first time to post – geektobe Oct 09 '19 at 17:47
  • @ghadah32 I don't mean anything in particular, but if my hint is helpful, and you come up with a result, yet you still aren't sure if your calculation is right, you can write down your calculation in the question, tag this question with #proof-verification and hopefully people would be able to scrutinise it. –  Oct 09 '19 at 17:49

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Yes, more precisely, since gcd is associative we have

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

Bill Dubuque
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