First slide

Permutations

Question

The number of integral solutions of $x^{2} - y^{2} = 352706$ is

Moderate

A

276
B

0
C

720
D

infinite

Solution

As 352706 is even, either both $x$ and $y$ are even or both $x$ and $y$ are odd.
If $x$ , $y$ are both even, $x^{2} - y^{2}$ is divisble by 4, but 352706 is not divisible by 4.
If $x$ , $y$ are both odd, $x^{2} - y^{2}$ is divisble by 4, but 352706 is not divisible by 4.

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