Relations between indices of Calabi--Yau varieties and pairs

We show that for any smooth Calabi--Yau variety, its index can be realized as the index of a Kawamata log terminal (klt) Calabi--Yau pair of lower dimension with standard coefficients. Our approach is based on an inductive argument on the dimension using the Beauville--Bogomolov decomposition. A key step in the argument is to prove that for $n\ge3$, any positive integer $m$ satisfying $\varphi(m)\le 2n$ can be realized as the index of a klt Calabi--Yau pair of dimension $n-1$.