## Maximum product of parts

### Problem 183

Published on Friday, 22nd February 2008, 05:00 pm; Solved by 3041; Difficulty rating: 45%Let N be a positive integer and let N be split into `k` equal parts, `r` = N/`k`, so that N = `r` + `r` + ... + `r`.

Let P be the product of these parts, P = `r` × `r` × ... × `r` = `r`^{k}.

For example, if 11 is split into five equal parts, 11 = 2.2 + 2.2 + 2.2 + 2.2 + 2.2, then P = 2.2^{5} = 51.53632.

Let M(N) = P_{max} for a given value of N.

It turns out that the maximum for N = 11 is found by splitting eleven into four equal parts which leads to P_{max} = (11/4)^{4}; that is, M(11) = 14641/256 = 57.19140625, which is a terminating decimal.

However, for N = 8 the maximum is achieved by splitting it into three equal parts, so M(8) = 512/27, which is a non-terminating decimal.

Let D(N) = N if M(N) is a non-terminating decimal and D(N) = -N if M(N) is a terminating decimal.

For example, ΣD(N) for 5 ≤ N ≤ 100 is 2438.

Find ΣD(N) for 5 ≤ N ≤ 10000.