Is this proof even valid? Is it true that all odd numbers can be uniquely expressed in the form $2^mq$? I saw this in a textbook I'm using to Self-study the Pigeon-hole principle and I'm quite sceptical about the claim in the bosses part of the text. The author claims the first 50 odd numbers can be uniquely expressed in the form $2^mq$, at first I taught it was a typo(as in, it should be of the form $2mq$ instead) but even it that were the case, it certainly wouldn't work for primes. So I need someone to help clear this up to me. !enter image description here

Yes, it is true that every natural number $n$ can be written in one and only one way as $2^mq$ where $q$ is an odd number. Let $2^m$ be the greatest divisor of $n$ which is a power of $2$ (note that $1$ is a power of $2$, since $1=2^0$). Now let $q=\frac n{2^m}$. It is clear that $q$ is odd; otherwise, $2^m$ wouldn't be the greatest power of $2$ that divides $n$. Finally, it is clear that $m$ is unique, since it was the defined as the greatest exponent of $2$ for which $2^m\mid n$.