Given \(a_i \in \mathbb{Z}^+ \forall 1\le i \le n\), and

\(a_1 + a_2 + \cdots + a_n = m\), where \(m\) is given positive integer and

\(a_1 a_2 \cdots a_n\) is maximum.

Find \(a_i, n\) for a given integer \(m\).

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## Comments

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TopNewestNote that \(2(k-2) > k\) for \(k > 4\), that \(k+1 > k\times1\) for \(k \ge 1\), and that \(3^2 > 2^3\).

Firstly, note that there are finitely many ways of writing \(m\) as a sum of positive integer (there are at most \(m\) integers in the sum, and each is no greater than \(m\)). Thus there certainly is a maximum possible product over all such sums (since there are only finitely many products to consider).

Thus the decomposition of \(m\) that yields the maximum product contains only the numbers \(2\) and \(3\). Since \(2^3 < 3^2\) and \(2+2+2=3+3\), any set of three \(2\)s can be replaced by two \(3\)s, resulting in a larger product. Thus the decomposition of \(m\) that yields the maximum product cannot have more than two \(2\)s.

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@Pi Han Goh @Mark Hennings

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