38873

The possibilities of the 5 digit set is:
{OOOOO}
{OEEEE}
{OOEEE}
{OOOEE}
{OOOOE}
{EEEEE}

Where O and E denote Odd and even integers
For possibilities 2 thru 5 there exists an multiple of 3 because the subset OOE or EEO is present in all sets but 1 & 6.

For Example: {4,7,14,5,3}
4 + 14 + 3 = 21|3 Therefore Set EEO

For sets 1 & 6 ({EEEEE} and {OOOOO})
If 1 of the number in the set is a multiple of 3 then any two other odd number will form a multiple of three. Finally if the difference of two of the numbers is a multiple of three then those and another number is also a multiple of three.
Therefore there are no possible solutions.