Problem 4

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

1976

Step 4 of 6: Remove repeated

1

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In plain words

The number $1$ contributes sum but no multiplicative growth, so every appearance of it can be profitably absorbed into nearby $2$ 's or $3$ 's.

1+1=2<1\cdot1\text{ is false; rather }1\cdot1<2

Detailed analysis

Two $1$ 's can be replaced by one $2$ : the sum stays $2$ , while the product changes from $1\cdot1=1$ to $2$ , which is larger. If one $1$ remains, then because $1976\equiv2\pmod3$ , there must be at least one $2$ ; replacing $1+2$ by $3$ preserves the sum and increases the product from $2$ to $3$ . Hence an optimizer uses only $2$ 's and $3$ 's.