How can I prove that it is not a perfect square?

Question

1. A $10$-digit number has one $1$, two $2$'s, three $3$'s and four $4$'s as its digits in some order. Prove that it can never be a perfect square.

Answer 1

The digit sum $30$ is divisible by $3$ but not by $9$. So our number is divisible by $3$ but not by $9$, and therefore cannot be a perfect square.