A consequence of the Fundamental Theorem of Arithmetic. The book said that:"If $S={1,2,3,.....,200}$, then for each $x \in S$, we may write $x=2^{k}y$, with $k \geq 0$, and gcd(2,y)=1." and the book added that this result follows from the Fundamental Theorem of Arithmetic, but I do not know how?
could anyone explain this for me?   

The fundamental theorem of arithmetic states that $x=2^a\times\prod p_i^{a_i}$.
Where $a$ is an integer eventually equal to $0$ and $p_i$ are prime numbers differents of $2$ and $a_i$ are integers $>0$.
So $p_i$ are odd since they are prime and different of $2$ so each $p_i$ is coprime with $2$, so $\prod p_i^{a_i}$ is coprime with $2$. So you have your result if you pick $y=\prod p_i^{a_i}$.
