Problem 299

If $N$ is divisible by both 5 and 11, then $N$ is divisible by $5 \times 11 = 55$.

This is because 5 and 11 have no common divisor larger than 1.

Therefore, we are looking for a multiple of 55 between 400 and 600 that is odd.

One way to find such a multiple is to start with a known multiple of 55, such as 550.

We can add or subtract 55 from this multiple and still obtain multiples of 55.

Note that $550 + 55 = 605$, which is too large.

Now $550 - 55 = 495$ which is in the correct range and is odd.

Since we are told that there is only such such integer, then it must be the case that $N = 495$.

The sum of the digits of $N$ is $4+9+5=18$.