We now have more than 70 problems under our belt and ready to attack the next one. Not that I really care about the number of problems, but this one is really really fun I think. It is easy to understand and have a really beautiful solution based on simple algebra. But lets get on with it, the problem reads

**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.

If we just wanted to search all the proper fractions in the search space we would have to search in the ball park of not something I am dying to do. Of course we could stop once we get about 3/7 and that would about half the search space, but still. So let’s look at another method. Continue reading →