Show that every positive integer n has a unique expression of the form n=2rm,r greater or equal 0, m a postive odd integer
From the "Unique-Prime-Factorization Theorem" we can write $\displaystyle N=p_1^{k_1}p_2^{k_2}..p_n^{k_n}$. Now let r be a number such that $\displaystyle 2^r \mid N$ but $\displaystyle 2^{r+1} \not ,\mid N$ and let $\displaystyle p_1=2$. Then $\displaystyle N=2^{r}m$, where $\displaystyle m=p_2^{k_2}..p_n^{k_n}$ and m is odd from the construction of r. This is a unique expression from the "Unique-Prime-Factorization Theorem".