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- Feb 13th 2010, 06:23 AM #1

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- Feb 13th 2010, 10:15 AM #2

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The first part, at least, is very straight forward:

$\displaystyle 0^2= 0 (mod 8)$, $\displaystyle 1^2= 1 (mod 8)$, $\displaystyle 2^2= 4 (mod 8)$, $\displaystyle 3^2= 9= 1 (mod 8)$, $\displaystyle 4^2= 16= 0 (mod 8)$, $\displaystyle 5^2= 25= 1 (mod 8)$, [tex] 6^2= 36= 4 (mod 8), and $\displaystyle 7^2= 49= 1 (mod 8)$. Since every other number is congruent to one of those mod 8, you are done.

For the second part just show that none of the ways of adding three such numbers is congruent to 7 mod 8.

- Feb 13th 2010, 12:06 PM #3

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