Question

The number of all possible pairs of integers $(m,n)$ which satisfy the equation $m^2+2m-35=2^n$ is

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is B $2$

$m^2+2m-35=2^n$
$\Rightarrow (m+7)(m-5)=2^n$
We observe that $m+7$ and $m-5$ are $12$ apart, and also observe that the only powers of $2$ which differ by $12$ are $4$ and $16.$
So, $n=2+4=6$
There are two cases :
$\left(I\right)$ $m+7=16$ and $m-5=4,$ which gives us $m=9$
$\left(II\right)$ $m+7=-4$ and $m-5=-16,$ which gives us $m=-11$
$\therefore$ The required pairs are $(9,6),(-11,6).$