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

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Re: If n is a positive integer, what is the remainder when 2n is divided [#permalink]
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Bunuel wrote:
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.

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.

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Re: If n is a positive integer, what is the remainder when 2n is divided [#permalink]
Bunuel wrote:
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.

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
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If n is a positive integer, what is the remainder when 2n is divided [#permalink]
Bunuel wrote:
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.

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.

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Re: If n is a positive integer, what is the remainder when 2n is divided [#permalink]
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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.
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If n is a positive integer, what is the remainder when 2n is divided [#permalink]
Bunuel wrote:
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.

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

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If n is a positive integer, what is the remainder when 2n is divided [#permalink]
GMATinsight wrote:
Bunuel wrote:
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.

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. This is incorrect

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

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
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Re: If n is a positive integer, what is the remainder when 2n is divided [#permalink]
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

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Re: If n is a positive integer, what is the remainder when 2n is divided [#permalink]
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Re: If n is a positive integer, what is the remainder when 2n is divided [#permalink]
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