Question

Prove that every positive integer different from $1$ can be expressed as a product of a non-negative power of $2$ and an odd number.

Answer 1

Let $n$ be positive integer other then $1$. By the fundamental theorem of Arithmetic $n$ can be uniquely expressed as power of primes gin ascending order so, let
$n={p_1}^{a_1}{p_2}^{a_2}...{p_k}^{a_k}$ can be the unique factorisation of $n$ into primes with $p_1<p_2<p_3...<p_k$.
Clearly, either $p_1=2$ and $p_2,p_3...<p_k$ are odd positive integers or each of $p_1,p_2...,p_k$ is an odd positive integer.
Therefore, we have the following cases:

Case $\left(I\right)$ When $p_1=2$ and $p_2,p_3...,p_k$ are odd positive integers: In this case, we have

$n=2^{a_1}{p_2}^{a_2}{p_3}^{a_3}...{p_k}^{a_k}$
$\Rightarrow$ $n=2^{a_1}\times ({p_2}^{a_2}{p_3}^{a_3}...{p_k}^{a_k})$ An odd positive integer.
$\Rightarrow$ $n=2^{a_1}\times$ An odd positive integer.
$\Rightarrow$ $n=$ (A non-negative power of $2$) $\times$ (An odd positive integer)

Case $\left(II\right)$ When each of $p_1,p_2,p_3,...,p_k$ is an odd positive integer:
In this case, we have
$n={p_1}^{a_1}{p_2}^{a_2}{p_3}^{a_3}...{p_k}^{a_k}$
$\Rightarrow$ $n=2^0\times ({p_1}^{a_1}{p_2}^{a_2}{p_3}^{a_3}...{p_k}^{a_k})$
$\Rightarrow$ $n=$ (A non-negative power of $2$) $\times$ (An odd positive integer)

Hence, in either case $n$ is expressible as the product of a non-negative power of $2$ and positive integer.