I need help on the following...

Show that every positive integer n has a unique expression of the form n=(2^r)m, r>=0, m is a positve odd integer.

I know that we need to find the existence of n, then find the uniqueness.. need help...

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- Feb 13th 2009, 10:55 PM #1

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- Feb 2009
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## Ivan Nivens problem

I need help on the following...

Show that every positive integer n has a unique expression of the form n=(2^r)m, r>=0, m is a positve odd integer.

I know that we need to find the existence of n, then find the uniqueness.. need help...

- Feb 14th 2009, 08:19 AM #2
Existence is easily proven by strong induction.

1 is of that form as so is

Let and assume that this is true for all integers .

If is odd, then

If is even, then for some .

By the inductive hypothesis, – which proves the result by strong induction.

For uniqueness, suppose

Then since Similarly

and so and