Question

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

Complete step by step solution:
Option (A): Suppose \[w = 2\] . Then we get
 \[2 \otimes 1 = \dfrac{{{2^{2 + 1}}}}{{{2^{2 - 1}}}} = 4\] .
 \[ \Rightarrow \] \[w\] 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
 \[2 \otimes 1 = \dfrac{{{2^{2 + 1}}}}{{{2^{2 - 1}}}} = 4\] .
Similarly suppose \[w = 3\] . Then we get
 \[3 \otimes 1 = \dfrac{{{2^{3 + 1}}}}{{{2^{3 - 1}}}} = 4\] .
 \[ \Rightarrow \] \[w\] 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
 \[2 \otimes 1 = \dfrac{{{2^{2 + 1}}}}{{{2^{2 - 1}}}} = 4\] .
Similarly suppose \[w = 3\] . Then we get
 \[3 \otimes 1 = \dfrac{{{2^{3 + 1}}}}{{{2^{3 - 1}}}} = 4\] .

Similarly suppose \[w = 4\] . Then we get
 \[4 \otimes 1 = \dfrac{{{2^{4 + 1}}}}{{{2^{4 - 1}}}} = 4\] .
 \[ \Rightarrow \] \[w\] 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
 \[k \otimes 1 = \dfrac{{{2^{k + 1}}}}{{{2^{k - 1}}}} = 4\] .
 \[ \Rightarrow \] \[w\] 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
 \[k \otimes 1 = \dfrac{{{2^{k + 1}}}}{{{2^{k - 1}}}} = 4\] .
 \[ \Rightarrow \] \[w\] 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.