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If n is odd, we can write it as [tex]\displaystyle \begin{align*} n = n \cdot 2^0 \end{align*}[/tex].Prove that every n E N can be written as a product of odd integer and a non-negative integer power of 2.
For instance: 36 = 22 * 9
This is not enough to prove the required claim.If n is odd, we can write it as [tex]\displaystyle \begin{align*} n = n \cdot 2^0 \end{align*}[/tex].
If n is even (including 0), it must have a factor of 2, so we can write it as [tex]\displaystyle \begin{align*} n = 2^1 k \end{align*}[/tex].
What about powers of 2? 2^k (k being a positive integer) has no odd factors, unless you want to include 1.Prove that every n E N can be written as a product of odd integer and a non-negative integer power of 2.
For instance: 36 = 22 * 9
Well, we want to include 1. Thus, if $k$ is a nonnegative integer, then $2^k=1\cdot2^k$: here 1 is an odd integer and $2^k$ is a non-negative integer power of 2, so this factorization satisfies the problem statement.What about powers of 2? 2^k (k being a positive integer) has no odd factors, unless you want to include 1.