$(\text{even}/\text{even})\times \text{even} = \text{odd}$?

Question

Do even natural numbers $e_1,e_2,e_3$ exist with $\frac{e_1}{e_2}\cdot e_3=o$, such that $o$ is an odd natural number?

If exist do they have any relation for this even tuple$(e_1,e_2,e_3)$?

Answer 1

If $v(n)$ denotes the exponent in the power of $2$ in the prime factorization of the even number $n$ ,

then $\frac{e_1}{e_2}\cdot e_3$, if it is an integer, is odd if and only if $v(e_1)+v(e_3)=v(e_2)$.

The easiest example is $\frac{2}{4}\cdot 2$

Answer 2

Here's a simple counterexample: (4/8)*6=3.

Answer 3

If $e_1$, $e_2$ and $e_3$ are even and $o$ is odd, then:$$(e_1/ e_2)e_3=o\iff e_1\cdot e_3=o\cdot e_2\implies e_2\equiv0\bmod 4.$$

On the other hand, if $o$ is odd and $e_2\equiv 0\bmod 4$, then: $$o\cdot e_2\equiv 0\bmod 4\implies \exists e_1,e_3\text{ even, such that } o\cdot e_2=e_1\cdot e_3\iff (e_1/e_2)e_3=o.$$