# Unique expression of an integer

From the "Unique-Prime-Factorization Theorem" we can write $N=p_1^{k_1}p_2^{k_2}..p_n^{k_n}$. Now let r be a number such that $2^r \mid N$ but $2^{r+1} \not ,\mid N$ and let $p_1=2$. Then $N=2^{r}m$, where $m=p_2^{k_2}..p_n^{k_n}$ and m is odd from the construction of r. This is a unique expression from the "Unique-Prime-Factorization Theorem".