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1969 Canadian MO Problems/Problem 7

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Problem

Show that there are no integers $\displaystyle a,b,c$ for which $\displaystyle a^2+b^2-8c=6$.

Solution

Note that all perfect squares are equivilant to $\displaystyle 0,1,4\pmod8.$ Hence, we have $\displaystyle a^2+b^2\equiv 6\pmod8.$ It's impossible to obtain a sum of $\displaystyle 6$ with two of $\displaystyle 0,1,4,$ so our proof is complete.

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