Problem 4

Determine, with proof, the greatest number which is the product of positive integers whose sum is

1976

Step 5 of 6: Use as many

3

's as possible

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In plain words

Three small factors of $2$ are less efficient than two factors of $3$ for the same total sum; this is the discrete version of the “near $e$ ” product heuristic.

2+2+2=3+3,\quad 2^3=8<9=3^2

Detailed analysis

Whenever three $2$ 's occur, replace them by two $3$ 's. The sum remains $6$ , but the product increases from $2^3=8$ to $3^2=9$ . Therefore an optimizer has at most two $2$ 's.