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No matter what 5 integers you choose, there must be 3 integers that add up to a multiple of 3. To do this, first calculate the integers mod 3. There are 4 types of situations where 3 integers add up to 0 mod 3: 0+0+0, 1+1+1, 2+2+2, 0+1+2.
The combination of 5 numbers can be separated into 2 categories after mod 3:
First, if there is at least 3 of a kind, there must be 3 integers that add up to a multiple of 3 because any 3 of a kind sums up to a multiple of 3.
Secondly, if there is no 3 of a kind, the combination of the remainders must be 2, 2, 1, in any order. In that case,there must be 3 integers that add up to a multiple of 3 since 0+1+2 sums up to a multiple of 3.
Therefore, there must be 3 integers that add up to a multiple of 3 no matter what 5 integers you choose.