mathschallenge.net

Problem

A fraction whose numerator (top number) is less than its denominator (bottom number) is called a simple fraction.

1 9

2 9

4 9

5 9

7 9

8 9

There are six simple fractions involving ninths that cannot be cancelled down. How many simple fractions with a denominator equal to 24 cannot be cancelled down?

Solution

We are looking for the set of integers less than 24 that are co-prime (have no factors in common) with 24: 1, 5, 7, 11, 13, 17, 19 and 23.
So there are 8 simple fractions with a denominator of 24 that cannot be cancelled down.

Investigate difference fractions.
(Hint: look at the numbers that are not co-prime to begin with.)