## 14. Square

Can the square of an integer end with three identical non-zero digits?
14. First, note that the square of any integer must end in one of the following digits: $0,1,4,5,6$ or 9. Moreover, the square of any integer is either divisible by 4 (if the number is even) or equal to a number divisible by 4 increased by 1 (if the original number is odd). However, numbers ending in 11, 55, 66, or 99 are not divisible by 4 and give remainders of $3,3,2$ and 3, respectively, when divided by 4, so they cannot be perfect squares. We are left to investigate the cases of numbers ending in 44, and more specifically, in 444. It is not difficult to establish that although 444 itself is not a perfect square, $1444=$ $=(38)^{2}$.

Therefore, integers whose squares end in three identical non-zero digits do exist.