Question

A perfect square is an integer that is the square of an integer. Suppose that $m$ and $n$ are positive integers, such that $mn>15$. If $15mn$ is a perfect square, calculate the least possible value of $mn$.

• A. $60$
• B. $80$
• C. $70$
• D. $40$
• E. $50$

Answer 1

Answer: (A)

$60$

As given $15mn$ is a perfect square.
So, $\sqrt{15mn}$ to be an integer.
$\sqrt{15mn}=\sqrt{5\times 3\times m\times n}$
Assume $m=3$ , $n=5$
$\Rightarrow$ $\sqrt{5\times 3\times m\times n}$=$\sqrt{5^2{.3}^2}=15$
But it is $mn<15$.
So, assume $m$ $=3\times 2$ ; $n=5\times 2$
$\Rightarrow$ $\sqrt{5.3\times m\times n}$$=\sqrt{\mathrm{5.3.3.2.5.2}}$=$\sqrt{5^2{.3}^2{.2}^2}=30$ is an integer.
As you can see that the least possible value of $m$ and $n$ are $6$ and $10$.
Hence, option A is correct.