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## Project Euler 72: How many elements in a set of reduced proper fractions

#### Project Euler 72 Problem Description

Project Euler 72: Consider the fraction, n/d, where n and d are positive integers. If nd and GCD(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 there are 21 elements in this set.

How many elements would be contained in the set of reduced proper fractions for d ≤ 1,000,000?

#### Project Euler 72 Solution

Runs < 0.750 seconds in Python 2.7.

Use this link to get the Project Euler 72 Solution Python 2.7 source.

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## Discussion

### 3 Responses to “Project Euler 72 Solution”

1. Well, I love the method, without seing it wrote I could not find phi back.

Posted by deff | July 25, 2016, 6:53 AM
2. Thanks for this post. I was stuck on this problem – using brute force, calculating totients for each number up to 1MM. It took forever!

Posted by Robert Tsai | September 28, 2011, 7:08 AM
3. I can not help to say this method is great!

Posted by Xiekaixuan | September 20, 2011, 12:47 AM