Euler's Theorem: Euler's Totient Function

The fractions that cannot be reduced are those in which the numerator has no prime factors in common with 24, or equivalently, those in which the numerator is relatively prime with 24. Thus, the number of irreducible fractions is $\phi(24)$ , where $\phi$ is Euler's totient function . Since the prime factors of $24=2^3\times 3$ are 2 and 3 only, every integer that is a multiple of neither 2 nor 3 is relatively prime with 24.

Therefore, $\phi(24) =24 \left( 1 - \dfrac12\right) \left( 1 - \dfrac13\right) = 8,$ and thus 8 of the 24 fractions are irreducible.