What is smallest possible integer k such that $1575 \times k$ is perfect square? I wanted to know how to solve this question:


What is smallest possible integer $k$ such that $1575 \times k$ is a perfect square?
a) 7, b) 9, c) 15, d) 25, e) 63. The answer is 7.


Since this was a multiple choice question, I guess I could just put it in and test the values from the given options, but I wanted to know how to do it without hinting and testing. Any suggestions?

If $1575\times k$ is a perfect square, we can write it as
$\prod_k p_k^2$. Since $1$ is not among the choices given, let's assume that $1575$ is not a square from the beginning.
Therefore $9=3^2$ and $25=5^2$ are ruled out, since multiplying a non square with a square doesn't make it a square. $63=9\cdot 7$ is ruled out, since $7$ would be the smaller possible choice.
We are left with $7$ and $15$ and it jumps to the eye that $1575$ is dividible by $15$ twice: 
$$
1575=1500+75=100\cdot 15+5\cdot 15=105\cdot 15=(75+30)\cdot 15=(5+2)\cdot 15^2.
$$
So what are you left with...? Good luck ;-)
