Here is what i got so far

The product of us is an odd number.

this implies that every numbers are odd.

The sum of us is a cube number.

since the limit is 100 and that every numbers are odd, and adding an odd amount of odd numbers results in an odd number,
we can deduce that only only 1, 27, 125, 343 are possible sums. since every numbers must be different, 1 is impossible.

The product of us is a cube number.

I have no idea how to proceed other than brute force,
but only 1 combination of numbers add up to 27 : 1+3+5+7+11. And their product is not a cube number. Therefor, 27 is also an impossible sum.
which leaves 125 and 343 as possible sums.

Here is what I got so far

The product of us is an odd number.

this implies that every numbers are odd.

The sum of us is a cube number.

since the limit is 100 and that every numbers are odd, and adding an odd amount of odd numbers results in an odd number,
we can deduce that only only 1, 27, 125, 343 are possible sums. since every numbers must be different, 1 is impossible.

The product of us is a cube number.

I have no idea how to proceed other than brute force,
but only 1 combination of numbers add up to 27 : 1+3+5+7+11. And their product is not a cube number. Therefor, 27 is also an impossible sum.
which leaves 125 and 343 as possible sums.