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Maximum product of natural numbers if their sum is constant

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December 6, 2010
inequalitiesnumber theory unsolvednumber theory

Problem Statement

If n1,n2,⋯ ,nkn_1, n_2, \cdots, n_k are natural numbers and n1+n2+⋯+nk=nn_1+n_2+\cdots+n_k = n, show that max(n1n2⋯nk)=(t+1)rtk−r,max(n_1n_2\cdots n_k)=(t + 1)^rt^{k-r}, where t=[nk]t =\left[\frac{n}{k}\right] and rr is the remainder of nn upon division by kk; i.e., n=tk+r,0≤r≤k−1n = tk + r, 0 \le r \le k- 1.
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