Relatively Prime Counts and Euler’s Totient

Euler's totient $\phi(n)$ counts the positive integers less than $n$ that share no factor with $n$.

For a prime $p$, every one of $1, 2, \ldots, p-1$ qualifies, so $\phi(p) = p - 1$. For a prime power, $\phi(p^k) = p^k - p^{k-1}$.

In general, factor $n$ and multiply: $\phi(n) = n\prod\left(1 - \frac{1}{p}\right)$ over the distinct primes $p$ dividing $n$. So $\phi(36) = 36 \cdot \frac{1}{2} \cdot \frac{2}{3} = 12$.

Watch the wording: "greater than 1 and less than $n$" excludes $1$, so subtract one from $\phi(n)$.