Break the product down into factors summing to 100

Obviously none of the smaller integers will be unity. Then if we compare a series of 2s and 3s, it will soon be clear that 3s would be better: For example, 2x2x2 (total 6) produces 8, whereas 3x3 (total 6) produces 9.

Furthermore, 4x4x4 (total 12) would give 64, while 3x3x3x3 (total 12) would provide the higher amount 81.

So, our optimum series will be thirty two 3s (total 96) and two 2s (total 4), making a grand total of 100, as required.

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