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Thread: Unique expression of an integer

  1. #1
    Sep 2009

    Unique 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
    Last edited by mr fantastic; Sep 6th 2009 at 01:43 AM. Reason: Moved from another thread
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  2. #2
    Junior Member
    Aug 2009
    Gothenburg, Sweden
    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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