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 you take the prime factorisation of a perfect square, all the exponents come out even, because if $n=p_1^{a_1}\cdot p_2^{a_2} \cdots p_m^{a_m}$, then $n^2=p_1^{2a_1}\cdot p_2^{2a_2} \cdots p_m^{2a_m}$.  
Therefore, if you want to multiply $1575 = 3^2\cdot 5^2 \cdot 7$ by some number $k$ to make a perfect square, then the prime factorisation of $k$ will have to include an odd exponent for 7, and an even exponent for any other primes.  Clearly, the smallest such $k$ (apart from the trivial $k=0$) would have no primes other than 7 in its prime factorisation, and an exponent of 1 for the prime 7; in other words, $k=7$ is the smallest possible.
