Question

For all positive integers $$w$$ and $$y$$ , where $$w>y$$ , let the operation $$\otimes$$ be defined by
 $$w\otimes y={\displaystyle \frac{2^{w+y}}{2^{w-y}}}$$ . 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

Hint: We solve this question by considering each option. Since $$w>y$$ so we check $$w\otimes 1$$ equal to 4 or not for all values of $$w$$ such that $$w>1$$ by using $$w\otimes y={\displaystyle \frac{2^{w+y}}{2^{w-y}}}$$ .

Complete step by step solution:
Option (A): Suppose $$w=2$$ . Then we get
 $$2\otimes 1={\displaystyle \frac{2^{2+1}}{2^{2-1}}}=4$$ .
 $$\Rightarrow$$ $$w$$ has at least one positive integer such that $$w\otimes 1$$ equal to 4.
Option (A) is incorrect.
Option (B):
Suppose $$w=2$$ . Then we get
 $$2\otimes 1={\displaystyle \frac{2^{2+1}}{2^{2-1}}}=4$$ .
Similarly suppose $$w=3$$ . Then we get
 $$3\otimes 1={\displaystyle \frac{2^{3+1}}{2^{3-1}}}=4$$ .
 $$\Rightarrow$$ $$w$$ has at least two positive integers such that $$w\otimes 1$$ equal to 4.
 $$\Rightarrow$$ Option (B) is incorrect.
Option (C):
Suppose $$w=2$$ . Then we get
 $$2\otimes 1={\displaystyle \frac{2^{2+1}}{2^{2-1}}}=4$$ .
Similarly suppose $$w=3$$ . Then we get
 $$3\otimes 1={\displaystyle \frac{2^{3+1}}{2^{3-1}}}=4$$ .

Similarly suppose $$w=4$$ . Then we get
 $$4\otimes 1={\displaystyle \frac{2^{4+1}}{2^{4-1}}}=4$$ .
 $$\Rightarrow$$ $$w$$ has at least three positive integers such that $$w\otimes 1$$ equal to 4.
Option (C) is incorrect.
Option (D): Suppose $$w=k$$ , where $$k$$ is any positive integer such that $$k>1$$ then, we get
 $$k\otimes 1={\displaystyle \frac{2^{k+1}}{2^{k-1}}}=4$$ .
 $$\Rightarrow$$ $$w$$ has more than four positive integers such that $$w\otimes 1$$ equal to 4.
 Option (D) is incorrect.
Option (E): Suppose $$w=k$$ , where $$k$$ is any positive integer such that $$k>1$$ then, we get
 $$k\otimes 1={\displaystyle \frac{2^{k+1}}{2^{k-1}}}=4$$ .
 $$\Rightarrow$$ $$w$$ has more than four positive integers such that $$w\otimes 1$$ equal to 4.
Option (E) is correct.
So, the correct answer is “Option E”.

Note: Note that the given condition is important. Sometimes many people neglect and suffer to get the exact solutions. The number we are assuming has to satisfy a given condition else it cannot be considered.