Question

For all positive integers $w$ and $y$, where $w>y$, let the operation $\otimes$ be defined by $w\otimes y=\frac{2^{w+y}}{2^{w-y}}$. For how many positive integers $w$ is $w\otimes 1$ equal to $4$?

• A. None
• B. One
• C. Two
• D. Four
• E. More than four

Answer 1

Answer: (E)

More than four

GIven $w\otimes y=\frac{2^{w+y}}{2^{w-y}}$

When $y=1$

$\Rightarrow$ $w\otimes 1=\frac{2^{w+1}}{2^{w-1}}$

$\Rightarrow w\otimes y=\frac{2\times 2^w}{2^w/2}$

$\Rightarrow w\otimes y=\frac{4\times 2^w}{2^w}$

$\Rightarrow w\otimes y=4$

For any value of $w$ when $y=1$, the value of $w\otimes y=\frac{2^{w+y}}{2^{w-y}}$ is $4$.