If n is a positive integer, what is the remainder when 2n is divided

Question

If n is a positive integer, what is the remainder when 2n is divided by 8?

(1) n, when divided by 6, leaves remainder 5.
(2) 3n, when divided by 6, leaves remainder 3.

jedit · Answer

It should be  E

Statement 1: Insufficient

n = 6q + 5

This set has values  5, 11, 17, 23 ... These have remainder of  5, 3, 1, 7  respectively when divided by 8. If we multiply them by  2 , we get  10,22,34,46,  remainder when divided by 8 is 2,6,2,6 so not a singular value.

Statement 2: Insufficient

3n = 6r + 3

This set has values  3, 9, 15, 21, 27  and so on. Since this is  3n , n would be  1,3,5,7,9,11,13,15,17  and so on.  2n  would be  2,6,10,14,18,22  and so on. Remainder when  2n  is divided by  9  is,  2, 6, 2, 6 .

Statement 1+2:

In both set of values of n from statements 1 and 2, we see  5,11,17,23  as common.  2n  would be  10,22,34  and  46 . However remainder when divided by 8 is again not a singular value.

Luckisnoexcuse · Answer

2n = 8x + r ..r?

(1) n can take 5, 11, 17, 23, 29.....
2n becomes 10, 22, 34, 46, 58 
when divided by 8 r is 2, 6, 2, 6, 2

Not sufficient

(2) 3n takes 3, 9, 15, 21, 27, 33, 39, 45
n becomes 1, 3, 5, 7, 9, 11, 13, 15
2n becomes 2, 6, 10, 14, 18, 22, 26, 30
when divided by 8 r is 2, 6, 2, 6, 2, 6, 2
Not sufficient

On combining also we have two value of r (2 and 6)

Not Sufficient
E

ScottTargetTestPrep · Answer

We need to determine the remainder when 2n is divided by 8.

Statement One Alone:

n, when divided by 6, leaves remainder 5.

Thus, we see n can be a number such as 5, 11, 16, 21, 26, 31, 35, etc.

When 2(5) = 10 is divided by 8, the remainder is 2.

When 2(11) = 22 is divided by 8, the remainder is 6.

Statement one alone is not sufficient to answer the question.

Statement Two Alone:

3n, when divided by 6, leaves remainder 3.

Thus, we see that 3n can be a number such as 3, 9, 15, 21, 28, etc.

When 3n is 3, n is 1; when 3n is 9, n is 3; when 3n is 15, n is 5; etc.

In other words n will always be an odd number: 1, 3, 5, 7, ...

When 2(1) = 2 is divided by 8, the remainder is 2.

When 2(3) = 6 is divided by 8, the remainder is 6.

Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using our two statements, we see the first value for n that satisfies both statements is 5. We also see that in statement two, n can be any odd number. So, another number that would match is n = 11.

When 2(5) = 10 is divided by 8, the remainder is 2.

When 2(11) = 22 is divided by 8, the remainder is 6.

We see that the statements together are still not sufficient to answer the question.

Answer: E

Kinshook · Answer

Asked: If n is a positive integer, what is the remainder when 2n is divided by 8?

(1) n, when divided by 6, leaves remainder 5.
n = 6k + 5
2n = 12k +10
The remainder when 2n is divided by 8 = {6, 2}
NOT SUFFICIENT

(2) 3n, when divided by 6, leaves remainder 3.
3n = 6k + 3
n = 2k + 1
2n = 4k + 2
The remainder when 2n is divided by 8 = {6,2}
NOT SUFFICIENT

(1) + (2) 
(1) n, when divided by 6, leaves remainder 5.
n = 6k + 5
2n = 12k +10
The remainder when 2n is divided by 8 = {6, 2}
(2) 3n, when divided by 6, leaves remainder 3.
3n = 6k + 3
n = 2k + 1
2n = 4k + 2
The remainder when 2n is divided by 8 = {6,2}
Combining, we get 
The remainder when 2n is divided by 8 = {6,2}
NOT SUFFICIENT

IMO E

arvind910619 · Answer

Hmmm this question was a little tricky.

So basically the two statements are same

1) n = 6a + 5
2) 3n = 6b + 3

we can get statement 2 from 1.

3*n = 3*6a +3*5 
3n = 18a + 15 
3n = 18a + 12 + 3
3n = 6(3a+2) + 3 
3n = 6b + 3 where b=3a +2.

E is the answer

IanStewart · Answer

Remainders by 6 aren't really related to remainders by 8, so there's no reason to think the statements should be sufficient here. Using Statement 1, n can be 5 and 11, and those values also work with Statement 2. So using both Statements, n can be 5 and 11, and 2n can thus have a remainder of 2, when n=5, or 6, when n=11, when we divide by 8, and the answer is E.

GMATinsight · Answer

Question:  Remainder(\frac{2n}{8}) =  ?

Statement 1 : n, when divided by 6, leaves remainder 5.

i.e. n = 5, 11, 17, 23, 29... etc
i.e. 2n = 10, 22, 35, 46, 58... etc

i.e.  Remainder(\frac{2n}{8}) =2, 6, 3...  etc.

NOT SUFFICIENT

Statement 2 : 3n, when divided by 6, leaves remainder 3

i.e. 3n = 3, 9, 15, 21, 27, ... etc
i.e. n = 1, 3, 5, 7, 9, 11, 13, 15... etc
i.e. 2n = 2, 6, 10, 14, 18... etc

i.e.  Remainder(\frac{2n}{8}) =2, 6, 3...  etc.

NOT SUFFICIENT

COmbining teh statements

i.e. n = 5, 11, 17, 23, 29... etc
i.e. 2n = 10, 22, 35, 46, 58... etc

i.e.  Remainder(\frac{2n}{8}) =2, 6, 3...  etc.

NOT SUFFICIENT

Answer: Option E

Mo2men · Answer

There is a typo that affected your calculation as highlighted

when n =17, then 2n=34..............hence  Remainder(\frac{2n}{8}) =2...

There is no 3 at all in any reminder. It is all 2 &6

hiranmay · Answer

If n is a positive integer, what is the remainder when 2n is divided by 8?

(1) n, when divided by 6, leaves remainder 5 -->  insuff: n = 6p+5, if p =0, n = 5, 2n =10, so 2n divided by 8, reminder = 2, but if p =1, n = 11, 2n =22, so 2n divided by 8, reminder = 6 
(2) 3n, when divided by 6, leaves remainder 3-->  insuff: 3n = 6q+3 => n = 2q+1 (=2r+5), if q =0, n = 1, 2n =2, so 2n divided by 8, reminder = 2, but if q =1, n = 3, 2n =6, so 2n divided by 8, reminder = 6
Combining (1) & (2) we get,
n=6p+5, similar as (1), so not sufficient 
  Answer: E

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If n is a positive integer, what is the remainder when 2n is divided

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