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

Printable View

- Sep 6th 2009, 01:19 AMblazingmathUnique expression of an integer
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

- Sep 6th 2009, 01:58 AMDavidEriksson
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".