Problem 4

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

1976

Step 2 of 6: Eliminate every summand at least

5

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In plain words

Large parts are inefficient: splitting off a $3$ creates more multiplicative growth than leaving the large integer intact.

3(x-3)>x\quad\text{for }x\ge5

Detailed analysis

If a summand $x\ge5$ occurs, replace it by $3$ and $x-3$ . The sum is unchanged, while the product is multiplied by $3(x-3)/x>1$ because $3(x-3)>x$ . Thus an optimizer contains no summand $x\ge5$ .