Taking out factors
Number Theory
Level
4
Find the number of unordered pairs ( x, y ) for which the following equation is satisfied where x,y are integers.
The answer is 0.
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One can see that the number is divisible by 4, and not 8. Therefore of the three terms in (x^2+y^2)(x+y)(x–y), at least 1 term and at most 2 terms can be even.
Consider x and y both even; in that case, all 3 terms would be even. Consider that only one of x or y are even; in that case, none of the terms are even. Consider that both x and y are odd; in that case all 3 terms are even. Therefore no values of x and y can satisfy that equation. There's no need to factorize out the number completely.