2006 CEMC Fermat Problems/Problem 7

Problem

What is the smallest positive integer $p$ for which $\sqrt{2^3 \times 5 \times p}$ is an integer?

$\text{ (A) }\ 2 \qquad\text{ (B) }\ 5 \qquad\text{ (C) }\ 10 \qquad\text{ (D) }\ 1 \qquad\text{ (E) }\ 20$

Solution

For the square root to be an integer, $2^3 \times 5 \times p$ has to be a perfect square. This happens when the exponents in the prime factorization are all even.

If $p$ is an integer, this means that the smallest number this can happen for is $2^4 \times 5^2$ , where $p = 2 \times 5$ .

We then have:

$p = 2 \times 5 = \boxed {\textbf {(C) } 10}$

~anabel.disher

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2006 CEMC Fermat Problems/Problem 7

Problem

Solution