Find a positive integer smaller than 500 that has a remainder of 3 when divided by 5, a remainder of 6 when divided by 9, and a remainder of 8 when divided by 11.

Answer 1

The answer must appear on all these lists:
498, 493, 488, 483,... 13, 8, 3.
492, 483, 474, 465,... 24, 15, 6.
492, 481, 470, 459,... 30, 19, 8.

It must end in 8 or 3 to be on the first list.
Numbers in the second list that end in 8 or 3 are:
483, 438, 393, 348, 303, 258, 213, 168, 123, 78, 33.
Numbers on the third list that end in 3 or 8 are:
448, 393, 338, 283, 228, 173, 118, 63, 8

393 is the only number on all lists.

This is rather a "brute force" method, but it works.