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

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Given that both x and y are positive integers, and that  y = 3^{(x – 1)} – x , is y divisible by 6?

(1) x is a multiple of 3

(2) x is a multiple of 4

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CareerGeek · Accepted Answer

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

Archit3110 · Answer

#1
x is a multiple of 3
test with x= 3,6,9 we get yes and no ; insufficient
#2
x is a multiple of 4
x=4,8,12 .. we get no always ; sufficient
IMO B

Given that both x and y are positive integers, and that y=3^(x–1)–x, is y divisible by 6?

(1) x is a multiple of 3

(2) x is a multiple of 4

shameekv1989 · Answer

Given that both x and y are positive integers, and that  y=3^{(x–1)}–x , is y divisible by 6?

(1) x is a multiple of 3

(2) x is a multiple of 4

y=3^{(x–1)}–x  =>  y=\frac{3^{(x)}}{3}–x

For Y to be divisible by 6 it must have one 2 and one 3 in its prime factorization.

i.e. the question is asking whether  y=\frac{3^{(x)}}{18}–\frac{x}{6}  is integer

i) x is a multiple of 3 
First term in  y=\frac{3^{(x)}}{18}–\frac{x}{6}  will have one two 3's but we don't know if x has a 2 or not in its prime factorization. Same goes for the second term. 
- Insufficient

ii) x is a multiple of 4
First term in  y=\frac{3^{(x)}}{18}–\frac{x}{6}  has a 2 in its prime factorization and so has the second term but we don't know if x has a 3 or not in its prime factorization.

Combining i) and ii)
 y=\frac{3^{(x)}}{18}–\frac{x}{6}  x has both a 2 and a 3 in its prime factorization. First term will have two 3's and a 2 to cancel 18 and second term will have a 2 and a 3 to cancel out 6 and hence we will get y as an integer.

Answer - C

freedom128 · Answer

(1) x is a multiple of 3
If x=3, y=6 is divisible by 6.
If x=6, y= odd-even = odd is not divisible by 6=3*2.
NOT SUFFICIENT

(2) x is a multiple of 4
If so, y= odd-even = odd is NOT divisible by 6=3*2.
SUFFICIENT

FINAL ANSWER IS (B)

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exc4libur · Answer

(1) x is a multiple of 3   insufic

x=m(3): odd or even
if y=odd- =even, answer yes
if y=odd- =odd, answer no

(2) x is a multiple of 4   sufic

x=m(4)=even
if y=odd- =odd, answer no

Ans (B)

Jawad001 · Answer

Given that both x and y are positive integers, and that y=3^(x–1)-x, is y divisible by 6?

(1) x is a multiple of 3

(2) x is a multiple of 4

Solution:
The number divisible by 6 is also divisible by 2 and 3 and the number must be even number.
From statement (1), 
x = 3, 6, 9, 12, ……….
In the question, we are given, y=3^((x–1))-x
When, x =3, then y=3^(x–1)-x or, y = 3^(3–1))-3= 3^2-3 = 9 -3 = 6 = divisible by 6
When, x = 6, then y = 3^5-6= odd – even = odd = not divisible by 6
Not Sufficient.

From statement (2), x = 4, 8, 12, 16,……………… 
When x = 4, then y =3^3-4= odd = not divisible by 6.
When x = 8, then y = 3^7-8 = odd = not divisible by 6
Hence Sufficient.
Answer: B

unraveled · Answer

Given that both x and y are positive integers, and that  y=3^{x–1}–x , is y divisible by 6?

(1) x is a multiple of 3
Let x = 3k where k is positive integer > 0
 y=3^{x – 1} – x  =  3^{3k – 1} – 3k 
Since both the entities in y are multiple of 3, y is multiple of 3. It is only when k is even y would be odd and when k is odd y would be even.
If k= 1
 y=3^{3.1 – 1} – 3.1  =  y=3^2 – 3  = 6 YES.
If x = 2
 y=3^{3.2 – 1} – 3.2  =  y=3^5 – 6  = 237 NO.

INSUFFICIENT.

(2) x is a multiple of 4
Let x = 4k where k is positive integer > 0
 y=3^{x–1}–x  =  3^{4k – 1} – 4k 
Since  y=3^{4k – 1}  would always be odd and 4k would always be even, y would always be odd, a non multiple of 6
though  y = 3^{4k – 1} – 4k  would be multiple of 3 when k is a multiple of 3.

SUFFICIENT.

ANSWER B.

minustark · Answer

Given that both x and y are positive integers, and that  y=3(x–1)–x , is y divisible by 6?

(1) x is a multiple of 3

(2) x is a multiple of 4

1) since x is a multiple of 3, y =  3^(3k-1)  - 3k. Now when, k =1, y = 3^2 -3 = 6. but when k = 2, y = 3 ^5 - 6 = 3 (3^4 -2) = 3. (odd- even) = 3 * odd, so y will not be divisible by 6. not sufficient.

2) when x =4, y =  3^3  -4 = odd - even = odd, no odd integer can be divisible by 6. All multiples of 4 will be an even integer. In that case, we can write y = 3^ ODD - even = odd - even = odd integer, so in any case y is not divisible by 6. sufficient.

B is the answer.

lacktutor · Answer

Given that both x and y are positive integers, and that  y= 3^{x—1} —x , is y divisible by 6?

( Statement1 ): x is a multiple of 3
If x=3 , then y =  3^{3–1}—3= 9–3= 6  (Yes) 
If x= 6, then  y= 3^{6–1}—6= 243–6= 237  ( y is not divisible by 6 —NO) 
 Insufficient

( Statement2 ): x is a multiple of 4
—> x= 4a 
—> y must be divisible by 2 and 3 at the same time.
 y = 3^{4a—1} —4a  
—>  3^{4a—1}  is always odd 
—> 4a is always even 
Odd—Even = Odd 
Well, y can never be divisible by 2. 
—> y is not divisible by 6. (Always NO) 
 Sufficient

The answer is B
 
   Posted from my mobile device

ostrick5465 · Answer

Given that both x and y are positive integers, and that  y=3^(x–1)–x , is y divisible by 6?

(1) x is a multiple of 3
sometimes y divisible by 6, sometimes y indivisible by 6
 not suff 
(2) x is a multiple of 4
=> x = 4n
=>  y = 3^{4n-1} - 4n 
  3^{4n-1}  is odd, 4n is even
=> y is odd
=> y indivisible by 6
=> Suff
=>  Choice B

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

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

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