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What is the remainder when n is divided by 26, given that n divided by

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What is the remainder when n is divided by 26, given that n divided by [#permalink]

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What is the remainder when n is divided by 26, given that n divided by 13 gives “a” as the quotient and “b” as the remainder? (a, b and n are positive integers)

(1) a is odd
(2) b = 3
[Reveal] Spoiler: OA

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Re: What is the remainder when n is divided by 26, given that n divided by [#permalink]

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Bunuel wrote:
What is the remainder when n is divided by 26, given that n divided by 13 gives “a” as the quotient and “b” as the remainder? (a, b and n are positive integers)

(1) a is odd
(2) b = 3


Target question: What is the remainder when n is divided by 26

Given: n divided by 13 gives “a” as the quotient and “b” as the remainder? (a, b and n are positive integers)
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2

So, we can taken the given information and write: n = 13a + b

Statement 1: a is odd
There's no information about b, so it will be impossible to determine the remainder when divided by 26.
Consider these two cases:
Case a: a = 1 and b = 2. In this case, n = (13)(1) + 2 = 15, so n divided by 26 leaves a remainder of 15
Case b: a = 1 and b = 3. In this case, n = (13)(1) + 3 = 16, so n divided by 26 leaves a remainder of 16
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b = 3
Consider these two contradictory cases:
Case a: a = 2 and b = 3. In this case, n = (13)(2) + 3 = 29, so n divided by 26 leaves a remainder of 3
Case b: a = 1 and b = 3. In this case, n = (13)(1) + 3 = 16, so n divided by 26 leaves a remainder of 16
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that a is odd, which means a = 2k + 1 for some integer k
Statement 2 tells us that b = 3

So, let's take the given information (n = 13a + b) and replace a with 2k + 1 and replace b with 3 to get:
n = 13(2k + 1) + 3
= 26k + 13 + 3
= 26k + 16
Here we can see that n is 16 greater than some multiple of 26, so when we divide n by 26, the remainder will be 16
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent
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What is the remainder when n is divided by 26, given that n divided by [#permalink]

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Bunuel wrote:
What is the remainder when n is divided by 26, given that n divided by 13 gives “a” as the quotient and “b” as the remainder? (a, b and n are positive integers)

(1) a is odd
(2) b = 3

n = 13a + b = 13(a-1)+b +13
(1) a is odd
=> a - 1 is even
=> 13(a-1) is divisible by 26
=> When n divided by 26, the remainder is 13+b
We do not know the value of b so INSUFFICIENT
(2) b = 3
=> We do not know a is odd or even
A is odd: the remainder: 13+b
a is even: the remainder: b
Insufficient

(1) + (2) The remainder is 13+3 = 16
SUFFICIENT
Ans: C

Last edited by camlan1990 on 20 Oct 2015, 12:41, edited 1 time in total.

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Re: What is the remainder when n is divided by 26, given that n divided by [#permalink]

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New post 20 Oct 2015, 12:24
GMATPrepNow wrote:
Bunuel wrote:
What is the remainder when n is divided by 26, given that n divided by 13 gives “a” as the quotient and “b” as the remainder? (a, b and n are positive integers)

(1) a is odd
(2) b = 3


Target question: What is the remainder when n is divided by 26

Given: n divided by 13 gives “a” as the quotient and “b” as the remainder? (a, b and n are positive integers)
There's a nice rule that say, "If N divided by D equals Q with remainder R, then N = DQ + R"
For example, since 17 divided by 5 equals 3 with remainder 2, then we can write 17 = (5)(3) + 2

So, we can taken the given information and write: n = 13a + b

Statement 1: a is odd
There's no information about b, so it will be impossible to determine the remainder when divided by 26.
Consider these two cases:
Case a: a = 1 and b = 2. In this case, n = (13)(1) + 2 = 15, so n divided by 26 leaves a remainder of 15
Case b: a = 1 and b = 3. In this case, n = (13)(1) + 3 = 16, so n divided by 26 leaves a remainder of 16
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: b = 3
Consider these two contradictory cases:
Case a: a = 2 and b = 3. In this case, n = (13)(2) + 3 = 29, so n divided by 26 leaves a remainder of 3
Case b: a = 1 and b = 3. In this case, n = (13)(1) + 3 = 16, so n divided by 26 leaves a remainder of 16
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that a is odd, which means a = 2k + 1 for some integer k
Statement 2 tells us that b = 3

So, let's take the given information (n = 13a + b) and replace a with 2k + 1 and replace b with 3 to get:
n = 13(2k + 1) + 3
= 26k + 13 + 3
= 26k + 16
Here we can see that n is 16 greater than some multiple of 26, so when we divide n by 26, the remainder will be 16
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer = C

Cheers,
Brent


Hi Brent, actually I wanted to place 2k+1 for St1+ST2, but what is with the case if our odd number=1 ? Can we always insert 2k+1 for an odd number ?
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Re: What is the remainder when n is divided by 26, given that n divided by [#permalink]

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New post 20 Oct 2015, 12:31
BrainLab wrote:
Hi Brent, actually I wanted to place 2k+1 for St1+ST2, but what is with the case if our odd number=1 ? Can we always insert 2k+1 for an odd number ?


Yes, you can always insert 2k+1 for an odd integer. That is actually how we define an odd integer.

Cheers,
Brent
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Re: What is the remainder when n is divided by 26, given that n divided by [#permalink]

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New post 29 Jun 2016, 09:39
GMATPrepNow wrote:
BrainLab wrote:
Hi Brent, actually I wanted to place 2k+1 for St1+ST2, but what is with the case if our odd number=1 ? Can we always insert 2k+1 for an odd number ?


Yes, you can always insert 2k+1 for an odd integer. That is actually how we define an odd integer.

Cheers,
Brent



Hello!

wouldn't it be 2k-1? in that sense 13(2k-1) + 3 would not igual 16. I am confused about this because I read that the sequence of all positive odd integers is defined by An = 2N-1. Would this matter for the solution of the problem?

Thank you for your response.

Greetings.
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Re: What is the remainder when n is divided by 26, given that n divided by [#permalink]

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New post 29 Jun 2016, 10:06
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Ilomelin wrote:
GMATPrepNow wrote:
BrainLab wrote:
Hi Brent, actually I wanted to place 2k+1 for St1+ST2, but what is with the case if our odd number=1 ? Can we always insert 2k+1 for an odd number ?


Yes, you can always insert 2k+1 for an odd integer. That is actually how we define an odd integer.

Cheers,
Brent



Hello!

wouldn't it be 2k-1? in that sense 13(2k-1) + 3 would not igual 16. I am confused about this because I read that the sequence of all positive odd integers is defined by An = 2N-1. Would this matter for the solution of the problem?

Thank you for your response.

Greetings.


2k - 1 is also an odd number for all integers k. However, if we use 2k-1 here, we need to do a little extra work at the end.

When we use 2k - 1, we get: n = 13(2k - 1) + 3
Simplify to get: n = 26k - 10
In other words, n is 10 LESS than some multiple of 26
Hmmm, what does this tell us about the remainder when n is divided by 26?
To find out, notice that we can take n = 26k - 10 and rewrite is as n = 26(k - 1 + 1) - 10
Or..... n = 26(k - 1) + 26 - 10
Simplify to get: n = 26(k-1) +16
So, n is 16 GREATER than some multiple of 26
So, we when we divide n by 26, we get a remainder of 16

PRO TIP: use 2k + 1 when you need a nice generic odd number

Cheers,
Brent
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Re: What is the remainder when n is divided by 26, given that n divided by [#permalink]

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New post 29 Jun 2016, 12:32
Bunuel wrote:
What is the remainder when n is divided by 26, given that n divided by 13 gives “a” as the quotient and “b” as the remainder? (a, b and n are positive integers)

(1) a is odd
(2) b = 3


n= 13a +b

Statement 1= a is odd

There is no information about b

n= 13*1 +b
n= 13*3 +b
n= 13 *5 +b

Not sufficient.

Statement 2= b = 3
Multiple values possible.
n= 3
n= 16 (13*1 +3)
n= 29 (13*2+3)
n= 42 (13*3 +3)

Not sufficient.

Combining both statements:-
When a is 1 (odd) and b= 3, let's say 16

then when n/26 will have 16 as remainder

Similarly when n = 42 - 16 is remainder for n/26

C is the answer
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Re: What is the remainder when n is divided by 26, given that n divided by [#permalink]

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