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Question: Find the greatest common divisor of $-90$ and $100$.

Answer: The GCD of $-90$ and $100$ will also divide their sum, $10$. As $10$ divides $-90$ and $100$, the GCD of $-90$ and $100$ is $10$.

Why is it this true: "The GCD of $-90$ and $100$ will also divide their sum"?

It easy to see that $10$ divides $-90$ and $100$, but how do you know that the GCD of $-90$ and $100$ is $10$ and not a larger number?

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    $\begingroup$ What is it that you do not understand? Is it the first sentence "The GCD of -90 and 100 will also divide their sum, 10"? Is it the first part of the second sentence, "10 divides -90 and 100"? Or is it how these two connect together to the conclusion that "the GCD of -90 and 100 is 10"? Is it something else, like "GCD" itself? Be more specific, and you are more likely to get an answer that you 1) understand, and 2) find helpful. $\endgroup$ – Arthur Nov 15 '17 at 13:10
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The greatest common divisor between $2$ integers is the largest (in absolute value) integer that divides both integers.

If we consider two integers $a$ and $b$ and if we assume that $d$ is their greatest common divisor then there exists integers $m$ and $n$ such that $a = md$ and $b = nd$. This then implies that $d$ divides the sum $a+b$ since $a+b = md+nd = (m+n)d.$

In your case, the sum of $-90$ and $100$ is $10$. So, the greatest common divisor between $-90$ and $100$ must also divide $10$. The greatest divisor of $10$ is itself (as is the case with any integer up to absolute value) and so it is natural to check whether or not $10$ divides both $-90$ and $100$. If it divides both then we are done since $10$ is also the largest integer that divides $100+(-90)=10.$

Indeed, $10$ divides both $-90$ and $100$ and so it must be their greatest common divisor.

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Let $d$ be the GCD, then $d$ divides both $100$ and $-90$, so it has to divide the sum $10$, then $d$ divides $10$, so $d\le 10$, and also $10$ divides $100$ and $-90$, so $d\geq 10$, using both together $d=10$.

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