Prove or disprove: If the least common multiple of $a_1, a_2, ..., a_n$ is odd, then none of the $a_i$, $i = 1, ..., n$, are even.

Question

Here is what I imagine is a very simple question regarding divisibility, which is nevertheless giving me quite a bit of trouble (as is usually the case).

Let $a_1, a_2, ..., a_n$ be positive integers.

Prove or disprove: If the least common multiple of $a_1, a_2, ..., a_n$ is odd, then none of the $a_i$, $i = 1, ..., n$, are even.

Answer 1

Just to formalize the answer implicit in the comments, if $M$ is the least common multiple of $a_1,a_2,\cdots ,a_n$, then $p\mid a_i \Rightarrow p\mid M$ and $p\mid M \iff p|a_i$ for some $i$. In particular, this holds for $p=2$. Thus $2\not\mid M \Rightarrow 2\not \mid a_i$ for any $i$.