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

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

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

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

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.

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

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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

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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