if the lcm is simply the product, then the integers are pairwise prime

If $g:=\gcd(n_i,n_j)>1$ for some $i\neq j$.

Note that $\frac {n_1 \cdots n_k} {g} < n_1 \cdots n_k$ is a common multiplier of $n_1, \ldots ,n_k$, which implies $\text{lcm}(n_1, \ldots ,n_k)\leq\frac {n_1 \cdots n_k} {g}<n_1 \cdots n_k$

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Number Theory

Gcd And Lcm

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