  We show that the index of any smooth Calabi--Yau variety is the index of a
klt Calabi--Yau pair of lower dimension with standard coefficients.
Additionally, we show that any positive integer $m$ satisfying $\varphi(m)\le
2n$ is the index of a klt Calabi--Yau pair of dimension $n-1$ with standard
coefficients. This result is used in the proof of the first theorem.
