Torsion in abelian groups

The functor $F(M) = M\otimes \mathbb{Z}_m = M/mM$ is right-exact. Furthermore, if $M$ is finite, then $\#mM = \#(M/M[m])$ and $\#M[m] = \#F(M)$. The exact sequence \begin{align*} F(A) \to F(B) \to F(C) \to 1 \end{align*} then implies \begin{align*} \#F(C) = \frac{\# F(B)}{\# \ker F(B) \to F(C)} = \frac{\#F(B)}{\# \operatorname{im} F(A) \to \#F(B)} \geq \frac{\#F(B)}{\#F(A)}; \end{align*} that is, $\#F(B) \leq \#F(A) \#F(C)$.