Project Euler – Problem 71 Solution

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Problem

Consider the fraction, n/d, where n and d are positive integers. If n < d and HCF(n,d)=1, it is called a reduced proper fraction.

If we list the set of reduced proper fractions for d <= 8 in ascending order of size, we get:

1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8

It can be seen that 2/5 is the fraction immediately to the left of 3/7.

By listing the set of reduced proper fractions for d <= 1,000,000 in ascending order of size, find the numerator of the fraction immediately to the left of 3/7.

Solution

This problem is fairly easy, given that the answer we’re looking for much be very close to 3 / 7 (0.4285714286) I simply iterate through the denominators, d, and find the closest numerator, n, which will produce a value less than 3 / 7. Then filter the set so we end up with only the n, d pairs that have a GCD of 1 and pick the numerator from the n, d pair whose n / d fraction is the biggest.

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