# Show gcd(a,b) and gcd(a,c) are relatively prime [closed]

Let b and c ∈ Z. Suppose that b and c are relatively prime. Show that for all integers a, gcd(a, b) and gcd(a, c) are relatively prime.

## closed as off-topic by user147263, user223391, user99914, Leucippus, colormegoneNov 10 '15 at 3:42

This question appears to be off-topic. The users who voted to close gave this specific reason:

• "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." – Community, Community, Leucippus, colormegone
If this question can be reworded to fit the rules in the help center, please edit the question.

For contradiction, assume that exists an integer $a$ such that $\gcd(a,b),\gcd(a,c)$ has a common divisor $d\ge 2$. Then $d\mid \gcd(a,b)\implies d\mid a,b$ and $d\mid \gcd(a,c)\implies d\mid a,c$. But then $d\mid b,c$, so $\gcd(b,c)\ge d\ge 2$, contradiction.