Proof. Write $\exp(G)=\prod p_i^{k_i}$ Since $\exp(G)$ is the least common multiple of the orders of each group element, it follows that for each $i$ there is an element whose order is a multiple of $p_i^{k_i}$ say $\lvert c_i\rvert=a_i p_i^{k_i}$ Let $d_i=c_i^{a_i}$ Then $\lvert d_i\rvert=p_i^{k_i}$ The $d_i$ thus have pairwise relatively prime orders, and thus $$\left\lvert\prod d_i\right\rvert=\prod\left\lvert d_i\right\rvert=\exp(G)$$ so that $\prod d_i$ is the desired element.
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