Question

Suppose $m$ and $n$ are distinct integers. Can $\frac{3^m\times 2^n}{2^m\times 3^n}$ be an integer? Give reasons.

Answer 1

Given that: $\frac{3^m\times 2^n}{2^m\times 3^n}$

( m and n are distinct integers)

$=\frac{3^{(m-n)}}{2^{(m-n)}}$ ........Using [$a^{-m}=\frac{1}{a^m}$]

$={\left(\frac{3}{2}\right)}^{m-n}$

this cannot be integer as $\frac{3}{2}=1.5$ so ${\left(1.5\right)}^{(m-n)}$ whether $m>n$ or $n>m$ in all cases result will be decimal. So $\frac{3^m\times 2^n}{2^m\times 3^n}$ is not an integer.