Solution given with question. If \[31z5\] is a multiple of 3, where z is a digit, what might be the values of z? Since \[31z5\] is a multiple of 3, its sum of digits \[3+1+z+5=9+z\] is a multiple of 3; so \[9+z\] is one of these numbers: 0, 3, 6, 9, 12, 15, 18, ... . But since z is a digit, it can only be that \[9+z=9\] or 12 or 15 or 18. Therefore, z = 0 or 3 or 6 or 9.
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