Algorithm to decompose a number into the product of an integer and a base two exponencial

Question

So i`ve been asked to code an algorithm that decomposes an integer into the product of a base two exponencial and some integer.Something like number = k.(2^n) , k and n being random integers with k restricted to being odd.

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My question is not about the programming aspect of this problem but rather the algebraic one as i have failed to came up with a standard procedure into finding one.The problem says thats the combination k.(2^n) is unique for every integer, so i`ve managed to find the combinations for certian numbers but it gets exponencially dificult as the number increases.

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For example, given the number 12 the only possible combination under this restrictions is 12 = 3(2^2)

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Please feel free to edit and improve anything you feel necessary as english is not my mother language

Answer 1

Start with $M$. If $M$ is odd you are done.

If $M$ is even, let $M_1 =\frac M{2}$ and you have $2^1*M_1$.

If $M_1$ is odd you are done. Otherwise let $M_2 = \frac {M_1}{2}$ and you have $2^2*M_2$.

Repeat. Each step you will have $2^j*M_j$ and so long as $M_j$ is even you will continue to $M_{j+1} = \frac {M_j}2$ and have $2^{j+1}M_{j+1}$.

Eventually this will end you reach an odd $M_k$ and have $2^kM_k$.

Answer 2

One simple way is to start with the $k$ you are given and $n=0$. Then divide $k$ by $2$ as many times as possible, incrementing $n$ each time.

while $k$ is even
$\quad k=k/2 $
$\quad n=n+1$