Given that both x and y are positive integers, and that y = 3^(x – 1)

Given, \(y = 3^{(x - 1)} - x\)
Asked: Is \(\frac{y}{6}\) = Integer ?

Method 1:

(1) \(x\) is a multiple of \(3\)
--> Possible values of \(x = 3, 6, 9 . . . \)

If \(x = 3\), \(y = 3^{(3 - 1)} - 3 = 3^2 - 3 = 6\) --> Divisible by \(6\)
If \(x = 6\), \(y = 3^{(6 - 1)} - 6 = 3^5 - 6 = 3(3^4 - 2) = 3*odd\) --> NOT Divisible by \(6\)
--> No Definite value --> Insufficient

(2) x is a multiple of 4
--> Possible values of \(x = 4, 8, 12 . . . \)

If \(x = 4\), \(y = 3^{(4 - 1)} - 4 = 3^3 - 4 = 23\) --> NOT Divisible by \(6\)
If \(x = 8\), \(y = 3^{(8 - 1)} - 8 = 3^7 - 8 \) --> NOT Divisible by \(6\)
If \(x = 12\), \(y = 3^{(12 - 1)} - 12 = 3^11 - 12 = 3(3^10 - 4) = 3*odd\) --> NOT Divisible by \(6\)
--> A Definite NO --> Sufficient

Method 2:

(1) \(x\) is a multiple of \(3\)
--> Possible values of \(x = 3a\), for any positive integer '\(a\)'
--> \(y = 3^{(3a - 1)} - 3a = 3(3^{(3a - 2)} - a)\)
If \(a\) = odd, y = \(3(3^{(3a - 2)} - a)\) = \(3\)(odd - odd) = \(3\)(even) --> Divisible by \(6\)
If \(a\) = even, y = \(3(3^{(3a - 2)} - a)\) = \(3\)(odd - even) = \(3\)(odd) --> NOT Divisible by \(6\)
--> No Definite value --> Insufficient

(2) x is a multiple of 4
--> Possible values of \(x = 4b\), for any positive integer '\(b\)'
--> \(y = 3^{(4b - 1)} - 4b\) = odd - even = odd ALWAYS --> NOT Divisible by \(6\)
--> A Definite NO --> Sufficient

Option B