Find the Number!

Let the $3$ -digit number be $\overline{XYZ} = 100X + 10Y + Z$ for single digit numbers $X$ , $Y$ , and $Z$ .

Since adding $36$ to the number make the tens and ones digit will swap with each other, $10Y + Z + 36 = 10Z + Y$ , which solves to $Z = Y + 4$ . Since $Y$ and $Z$ are single digits, the possibilities for $(Y, Z)$ are $(0, 4)$ , $(1, 5)$ , $(2, 6)$ , $(3, 7)$ , $(4, 8)$ , and $(5, 9)$ .

Since $100X + 10Y + Z$ is a multiple of $4$ , and $100X$ is a multiple of $4$ , $10Y + Z$ must also be a multiple of $4$ . The possibilites for $(Y, Z)$ are now further limited to $(0, 4)$ and $(4, 8)$ .

Since the reverse of the $3$ -digit number is $495$ greater than the number itself, $100X + 10Y + Z + 495 = 100Z + 10Y + X$ , which solves to $Z = X + 5$ , so $Z \geq 5$ . The possibilites for $(Y, Z)$ is now limited to $(4, 8)$ , which makes $X = 3$ , $Y = 4$ , and $Z = 8$ for the number $\boxed{348}$ .