Take Three from Five

Is it always possible, from any set of five positive whole numbers, to choose three numbers that add up to a multiple of 3?
For example, if you start with 4, 7, 11, 12 and 18, you can choose 4, 11 and 12 which add up to 27 (a multiple of 3).
Try some examples of your own.
If you think it is always possible can you explain why?

See the AIMSSEC Notes for Teachers and the NRICH poster.

This problem is adapted from the NRICH task Take Three from Five with permission of the University of Cambridge. All rights reserved.