12. Which is the smallest square that has 2016 as a factor?
A $42^{2}$
B $84^{2}$
C $168^{2}$
D $336^{2}$
E $2016^{2}$
SolUTION
C

An integer is a square if, and only if, the exponent of each prime in its prime factorization is even.
The prime factorization of 2016 is $2^{5} \times 3^{2} \times 7$.
If 2016 is a factor of a square, then the exponents of the primes 2,3 and 7 in the factorization of the square must be no smaller than their exponents in 2016.
Hence the smallest square of which 2016 is a factor has the prime factorization $2^{a} \times 3^{b} \times 7^{c}$, where $a, b$ are $c$ are the smallest even integers which are greater than or equal to 5,2 and 1 , respectively. Therefore $a=6, b=2$ and $c=2$. Hence, the smallest square is of which 2016 is a factor is $2^{6} \times 3^{2} \times 7^{2}$.
Now
$$
2^{6} \times 3^{2} \times 7^{2}=\left(2^{3} \times 3^{1} \times 7^{1}\right)^{2}=(8 \times 3 \times 7)^{2}=168^{2} .
$$

Therefore the smallest square of which 2016 is a factor is $168^{2}$.