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Difference between revisions of "Mock AIME 1 2010 Problems/Problem 1"

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*[[Mock AIME 1 2010 Problems]]
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*[[Mock AIME 1 2010 Problems/Problem 2|Followed by Problem 2]]
 

Latest revision as of 08:46, 2 August 2024

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.

See Also

Mock AIME 1 2010 (Problems, Source)
Preceded by
First Problem 		Followed by
Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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