Mock AIME 1 2010 Problems/Problem 1

Problem

Let $a_k = 7k + 4$ . Find the number of perfect squares among $\{a_1, a_2, \ldots, a_{2010}\}$ .

Solution

Since $n^2 = 7k+4$ , $n^2$ is congruent to $4\pmod 7$ . Taking the square root of both sides yields $n \equiv \pm 2 \pmod 7$ . The maximum of $a_k$ is $14,074$ , whose square root lies between $118$ and $119$ . So, $n \leq 118$ . 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 too small for $a_k$ . Thus, there are $16 + 17 = \boxed{033}$ solutions.

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Mock AIME 1 2010 Problems/Problem 1

Problem

Solution

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