Product of exponent of finite group and exponent of its Schur multiplier divides order of group

Statement
Suppose $$G$$ is a finite group. Denote by $$M(G)$$ the Schur multiplier of $$G$$. Then, the product (in the sense of multiplication of natural numbers) of the exponent of $$G$$ and the exponent of $$M(G)$$ divides the order of $$G$$.

Related facts

 * Cyclic implies Schur-trivial
 * Schur multiplier of finite group is finite and exponent of Schur multiplier divides group order