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I don't believe there are any counter examples that can be used for this (I think it is true). Could someone help me prove it?

I understand why it's true (if I was right about that), but the proof itself is a bit tricky.

Martin Sleziak
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bert
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4 Answers4

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On one hand, and since everything's positive here, $\;gcd(a,b)\le a,b\;$ , but on the other $\;a\mid a\;\;and\;\;a\mid b\;$ , so

$$a\le gcd(a,b)\le a\implies gcd(a,b)=a$$

DonAntonio
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  • Could you go into more detail about this? I don't fully understand what you wrote. – bert Nov 12 '13 at 19:50
  • since $;gcd(a,b);$ is the largest divisor of both $;a,b;$ the first inequality in my answer follows, and since $;a;$ is one of those divisors according to the given info, the leftmost inequality in the last line follows, too...so we get equality! – DonAntonio Nov 12 '13 at 19:52
  • More generally if $,a\mid b,$ then $,d\mid a,b\iff d\mid a,,$ so $,a,b,$ and $,a,$ have the same set $S$ of common divisors $,d,,$ hence the same greatest common divisor (= max $S$), i.e. apply the universal property characterizing the gcd. – Bill Dubuque Nov 30 '21 at 00:45
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If $a|b$, then there is some integer $n$ such that $b = na$. Now the Euclidian algorithm says that $$ \gcd(a, b) = \gcd(a, na) = \gcd(a, na - na) = \gcd(a, 0) = a $$

Arthur
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Hint: Use the fact that

$gcd(a,b)=d$ iff $gcd(\frac{a}{d},\frac{b}{d})=1$.

Note: This result can be proved using the theorem that states that gcd can be expressed as a linear combination of the numbers.

Swapnil Tripathi
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Suppose that $a,b>0$ and $a|b$.

  1. $a$ is a common divisor of $a,b$.
  2. If $c$ is also a common divisor of $a,b$, then $c|a$.

This is the usual definition of gcd.

vadim123
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  • Sorry, but where did the c come from? – bert Nov 12 '13 at 19:52
  • The definition I am using is that $gcd(a,b)$ is the greatest common divisor. That is, it is a common divisor, and it is greatest in the sense that all other common divisors divide it. – vadim123 Nov 12 '13 at 22:08