# Statement

et 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 = 2.25 = 51.53632.

Let M(N) = Pmax 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 Pmax = (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.

# Solution

To get the solution for this I used the help from Wolfram Alpha. I entered the formula 8^n / n^n and it says the maximum is located in 8 / $e$. That applies to any value so we translate that easily into code to determine the denominator.

Now, that we know the denominator we need to determine whether it is a terminating or non-terminating decimal. A terminating decimal is a fraction whose denominator's prime factors are only 2 and 5, this is easily verified.

from math import ceil, e, floor from fractions import gcd if __name__ == '__main__': result = 0 print(e) for n in range(5, 10001): denom = round(n / e) n_2 = n d = gcd(n_2, denom) while d > 1: n_2 /= d denom /= d d = gcd(n_2, denom) while denom % 2 == 0: denom /= 2 while denom % 5 == 0: denom /= 5 if denom == 1: result -= n else: result += n print("The result is:", result)

The Python file is available for download here.