## Project Euler 183: Maximum product of parts

#### Problem Description

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.

#### Project Euler 183 Solution

Runs < 0.010 seconds in Python 2.7.Use this link to get the Project Euler 183 Solution Python 2.7 source.

#### Afterthoughts

- The value of
`k`that maximizes`P`as M(N) (or P_{max}) can be calculated as N/`e`. Where`e`is 2.718281828, the base of the natural logarithm. - As for determining D(N): A fraction will terminate if and only if the denominator, k, has for prime divisors only 2 and 5, that is, if and only if the denominator has the form 2
^{n}5^{m}for some exponents n ≥ 0 and m ≥ 0. The number of decimal places until it terminates is the larger of n and m.

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