TITLE: Find all $n$ such that $\sqrt{5n+2}$ is an integer.
QUESTION [5 upvotes]: Here is my solution. There is no such $n$.
If $n$ is odd, then, then $5n+2 \equiv 7 \pmod {10}$.
Else, $5n+2 \equiv 2\pmod {10}$.
But, the quadratic residues of $10$ are only $0,1,4,9,6,5$. Therefore, the expression is never a perfect square.
Is it correct?

REPLY [2 votes]: For it to be an integer, $5n+2$ must be a perfect square, Now the last digit of $5n+2$ can only be either a $2$ or a $7$, None of which can be the last digits of perfect squares, hence $5n+2$ can never be a perfect square.