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A proof of this result was given, however I am having trouble understanding why the part in bold is true.

Let $A$ be a non-trivial finite abelian group with $|A|$ having distinct prime divisors $p_1,p_2,\ldots,p_n$. Then write $|A|=\prod_{i=1}^n p_i^{e_i}$ with all $e_i>0$, and for each $i$ write $r_i=|A|/p_i^{e_i}$. By Lagrange's theorem, the order of $a\in A$ divides $|A|$ and so the order of $r_i a$ divides $p_i^{e_i}$.

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Note that $r_ia$ is killed by $p_i^{e_i}$ since $p_i^{e_i}r_i a=|A|a=0$.

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