Mock AIME 1 2010 Problems/Problem 1

Since $n^2 = 7k+4$ , $n^2$ is congruent to $4$ mod $7$ . Taking the square root of both sides, we obtain $n$ is congruent to $\pm 2$ mod $7$ . Solving, $n$ is congruent to either $2$ or $5$ mod $7$ . The maximum of $a_k$ is $14074$ , of which the square root is between $118$ and $119$ . So, $n$ must be less than $118$ . The series of solutions for $n$ is $2, 9, 16...114$ and $5, 12, 19...117$ . However, $2$ does not work because $2^2 = 4$ , a value not possible for $a_k$ . Thus, there are $16 + 17$ different solutions for a total of $\boxed{33}$ solutions.

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Mock AIME 1 2010 Problems/Problem 1