Math CentralQuandaries & Queries


Question from Roger:

I am stuck with this problem: find the gcd(a^2+b^2, a+b), where a & b are relatively prime integers not both zero.

Hi Roger,

If it was just $\gcd(a^2 - b^{2}, a+b)$ it would be easy since $a^2 - b^2 = (a+b)(a-b)$ but I can't factor $a^2 + b^{2}.$

Whatever the value is of $\gcd(a^2 - b^{2}, a+b)$ it must be true for any allowable values of $a$ and $b.$ I would try some examples. Choose two relatively prime integers $a$ and $b,$ evaluate $a^2 + b^2$ and $a + b$ and find their gcd. Try it again for another pair of relatively prime integers. and again.

What did you learn?


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