Mock AIME 4 2006-2007 Problems/Problem 14

Problem

Let $x$ be the arithmetic mean of all positive integers $k<577$ such that

$k^4\equiv 144\pmod {577}$ .

Find the greatest integer less than or equal to $x$ .

Solution

We will assume that there is at least one solution, otherwise the answer would be undefined.

Using the binomial theorem it is obvious that $(577-k)^4 \equiv k^4 \pmod {577}$ . Thus the solutions come in pairs $\{k,577-k\}$ , and hence their average is $\dfrac{577}2 = 288.5$ , and the answer is $\boxed{288}$ .

(In this case, there are four solutions: $276$ , $277$ , $300$ , and $301$ .)

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Mock AIME 4 2006-2007 Problems/Problem 14

Problem

Solution