How to deduce that $\gcd(a, b) = \gcd(a, a+b)$ given that the common divisors of $a$ and $b$ are the same as the common divisors of $a$ and $a+b$?

Question

This homework question asks to first prove that the common divisors of $a$ and $b$ are the same as the common divisors of $a$ and $a+b$, which I have done, and then deduce from this result that $\gcd(a, b) = \gcd(a, a+b)$.

Let $c|a$ and $c|(a+b)$, therefore $c|b$. I have tried expanding $\gcd(a,a+b)$ in terms of multiples of $c$ and considering the case when $c = \gcd(a,a+b)$, but it gets me nowhere. Am I overlooking something obvious?

score 3 · Accepted Answer · answered Oct 27 '13 at 23:45

3

If the two sets of common divisors are the same, then the greatest common divisors are the same.

answered Oct 27 '13 at 23:45

André Nicolas

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How to deduce that $\gcd(a, b) = \gcd(a, a+b)$ given that the common divisors of $a$ and $b$ are the same as the common divisors of $a$ and $a+b$?

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