There is one odd integer $N$ between 400 and 600 that is divisible by both 5 and 11 . The sum of the digits of $N$ is
(A) 11
(B) 8
(C) 10
(D) 16
(E) 18
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 , which is even.

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 $4+9+5=18$.

ANSWER: (E)