Question

The number of three-digit number $M$ such that ${10}^{3}M+M$ is a perfect square is

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

The correct option is A $0$

${10}^{3}M+M=M({10}^3+1)$
$=1001M=7\cdot 11\cdot 13M$

If ${10}^{3}M+M$ is a perfect square, then $M$ divides all of $7,11,13$
Thus $M\ge 7\cdot 11\cdot 13=1001$
Therefore, no such three-digit number exists.