$($The binary number system.$)$ Take any nonnegative integer n$,$ and let bkbk$1...$ b$2$b$1$b$0$ be the

binary digits of n$.$

$($a$)$ Show that $($bkbk$1...$ b$2$b$1$b$00)$bin is equal to $2$n$.($In other words, sticking a $0$ on the end

of a binary string multiplies the result by $2.)$

$1$

$($b$)$ Show that $($bkbk$1...$ b$2$b$1)$bin is equal to

n

$2$

$.($In other words, removing the last digit

from a binary string is the same as dividing the result by $2$ and rounding down.$)$

$($c$)$ In computer science, we call the binary operation described in $($a$)$ a left shift by one bit.

Similarly, we call the binary operation in $($b$)$ a right shift by one bit.

Suppose that n is a number with the following property: when written in binary, if n is

first shifted right and then shifted left, the result is a prime number. What could n have

been originally, and why?