## Counting fractions

### Problem 72

Published on Friday, 18th June 2004, 06:00 pm; Solved by 12921Consider 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 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?