Expression of an Integer as a Power of 2 and an Odd Number (Chartrand Ex 5.4.2[a]) 
Let $n$ be a positive integer. Show that every integer $m$ with $ 1 \leq m \leq 2n $ can be expressed as $2^pk$, where $p$ is a nonnegative integer and $k$ is an odd integer with $1 \leq k < 2n$.

I wrote out some $m$ to try to conceive the proof. I observed: 
$\bbox[5px,border:1px solid grey]{\text{Case 1: $m$ odd}}$ Odd numbers $\neq 2p$, thus the only choice is to put $p = 0$ and $k = m$.
$\bbox[5px,border:1px solid grey]{\text{Case 2: $m$ even and power of 2}}$ Then $p$ can be determined by division or inspection to "square with" the power of $2$. This requires $k = 1$. Is an explicit formula for $p$ necessary?
$\bbox[5px,border:1px solid grey]{\text{Case 3: $m$ even and NOT A power of 2}}$ 
$\begin{array}{cc}
\\
\boxed{m = 6}: 6 = 2^1 \cdot 3 \qquad \qquad & \boxed{m = 10}: 10 = 2^1 \cdot 5 \qquad  \qquad & \boxed{m = 12}: 12 = 2^2 \cdot 3  \\
\boxed{m = 14}: 14 = 2 \cdot 7 \qquad \qquad & \boxed{m = 18}: 18 = 2^1 \cdot 9 \qquad  \qquad & \boxed{m = 20}: 20 = 2^2 \cdot 5\\
\boxed{m = 22}: 22 = 2 \cdot 11 \qquad \qquad & \boxed{m = 24}: 24 = 2^3 \cdot 3 \qquad  \qquad & \boxed{m = 26}: 26 = 2^1 \cdot 13
\end{array}$

Solution's Proof by Contradiction: $\color{#0073CF}{\Large{\mathbb{[}}}$Let $p$ be the greatest nonnegative integer
  such that $2^p | m. \color{#0073CF}{\Large{\text{]}}}$ $\color{red}{\Large{\text{[}}}$ Then $m= 2^pk$ for some positive integer $k$. Necessarily $k$ is odd,
  for otherwise this would contradicts the definition of $p$. $\color{red}{\Large{\text{]}}}$

$\Large{\text{1.}}$ Could someone please expound on the answer? It differs from my work above?
$\Large{\text{2.}}$ Is there a general formula/pattern for Case $3$? 
I referenced 1. Source: Exercise 5.42(a), P125 of Mathematical Proofs, 2nd ed. by Chartrand et al
 A: I am not sure, what you are looking for as the proof is fine, but here are some alternatives:


*

*You could use unique prime factoritzation, so for any $m$, there are $k_i\geq 0$, such that:
$$m=2^{k_0}\cdot 3^{k_1}\cdot 5^{k_2}\cdot \dots$$
Then $p=k_0$ and $k=3^{k_1}\cdot 5^{k_2}\cdot \dots$ does the job.

*In the case, where $m$ is even, but not a power of $2$, assume to the contrary, there are numbers not representable in that way. Then, by the well-ordering principle, there is a smallest one, which we denote by $n$. As $n$ is even, there is a $m\in\mathbb N$ with $2m=n$. Since $m<n$, $m$ has a representation $m=2^pk$, but then $n=2^{p+1}k$, a contradiction.

A: This is easy by induction one $m$. If $m$ is even then $m/2$ can be written as $2^qk$ with $k$ odd by induction, and writing $m=2^{q+1}k$ will do. If $m$ is odd writing $m=2^0m$ will do.
Unique factorization is not needed here since proving uniqueness of the expression is not asked for. It would be easy though to show uniqueness for this case "by hand" even without using unique factorization.
The only "subtle" point (mainly because it might be overlooked) is proving that $k<2n$. The argument only gives $k\leq2n$, but the latter inequality together with $k$ odd implies $k<2n$.
A: $p$ is the greatest integer such that $2^p$ divides $m$. 
This means that : 


*

*there exists an integer $k$ such that $m = 2^p k$

*for any integer $q$, if there is an integer $l$ such that $m = 2^q l$, then $p \ge q$.
Suppose the $k$ you get by 1. is even. Then $k = 2l$ for some integer $l$. Hence $m = 2^p(2l) = 2^{p+1}l$. Now apply 2. with $q = p+1$ : you get that $p \ge p+1$. This is a contradiction, so $k$ must be odd.
For example, let $m = 24$. It turns out that $24 = 2^3 * 3$, so $2^3$ divides $24$. $3$ is the greatest integer $p$ such that $2^p$ divides $m$ (you can indeed check that $16,32,64,128,\ldots$ don't divide $24$) and so we find that indeed, $3$ is odd and $24 = 2^3 * 3$ was the decomposition we wanted.
The solution gave you not an algebraic formula for $p$ (as you may have wanted), but a logical formula for $p$, which is the next best thing, as it tells you exactly which $p$ you have to take.
A: For any positive integer, $m$, there will be some non-negative integer, $r$ such that $m < 2^r$. This implies that there must be some non negative integer $k \le r$ such that $2^k \mid m$ and $2^{k+1} \nmid m$.$
For example $m = 96 \lt 128 = 2^7$ and $m = 2^5 \times 3$ where $5 \le 7$.
A: I personally found this problem easier to do if you consolidate case 2 and 3 into one case and then prove by contradiction:
Assume m is even and that m  cannot be written in the form $2^p \bullet k$ where $p$ is a nonnegative integer and k is an odd integer with $1 \le k \le 2n$. Thus, $m \neq 2^p \bullet k$. We can rewrite $2^p \bullet k = 2(2^{p-1} \bullet k)$. (Note that because m is even $p \ge 1$ and so $p-1 \ge 0$). This implies that  m is not even, which directly contradicts our assumption that m is even. Therefore, we must conclude that m can be written in the form $2^p \bullet k$ where $p$ is a nonnegative integer and k is an odd integer with $1 \le k \le 2n$. 
