Simple Fractions Symmetry
Given that k and n are positive integers, and kn, we shall call k/n a simple fraction if it cannot be cancelled.
For example, when n=9, there are exactly six simple fractions:
For a given denominator, n2, prove there will always be an even number of simple fractions.
For any given denominator, n, there will be n1 proper fractions:
1/n, 2/n, ... , (n1)/n.
If HCF(k,n)1, it follows that nk will also share the same common factor. For example, HCF(8,28)=4 and as both 8 and 20 divide by 4, 208=12 will also divide by 4. That is, if k/n cancels, (nk)/n will also cancel.
Using this idea, we can see how fractions will cancel in pairs. The exception is n being even, in which case there will be an odd number of proper fractions. However, the numerator of the middle fraction will be n/2, which will cancel.
For a given denominator, n, how many simple fractions exist?