Unique expression of an integer

**blazingmath** · September 6th 2009, 01:19 AM

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

**DavidEriksson** · September 6th 2009, 01:58 AM

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".

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