Number of six-digit squares using 1-6

Question

How many six-digit perfect squares can be formed using all of the numbers $1$,$2$,$3$,$4$,$5$,$6$ as digits?

a) $5$

b) $19$

c) $7$

d) none

I have been able to make few observations in this question but not sufficient to solve the question-

1. Squares of no.s ending in $5$,$6$ end in $5$,$6$ respectively.
2. It won't be practical to list all these down since we are dealing in 6-digit.
3. I could generally use a sense of permuting these digits but while finding a perfect square but that wouldn't really help here.
4. Squares are of the form $4k$ or $4k+1$ but this is not true for every value of $k$.

Answer 1

Hint: If $n$ is a $6$-digit number using each of the digits $1,2,3,4,5,6$ exactly once, what is $n \pmod{9}$? What are the possible residues of perfect squares $\pmod{9}$?