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Math Help - Number Theory

  1. #1
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    Number Theory

    let n in Z with n>0. prove that there exist k,m in Z with m odd such that n=2^km.
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  2. #2
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    n = 2^{k}m

    (1) If n is odd, then let k = 0 and m = n ( 2^{0} \times n = n)

    (2) If n is even, then factor out 2's one-by-one until you reach an odd number, in which case you'll set it to m. To illustrate: n = 2n_{1} = 2^{2}n_{2} = 2^{3}n_{3} = ... \quad n_{1}, n_{2}, ... \in \mathbb{Z}^{+} Eventually, some m_{i} will be odd and this will be your m - giving you n = 2^{i}m_{i}
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