GCD of $-90$ and $100$

Question

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?

Answer 1

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.

Answer 2

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 $d=10$.