n and k are positive integers. When n is divided by 23, the quotient i

Question

n and k are positive integers. When n is divided by 23, the quotient is 2k and the remainder is j. What is the value of j + k?

(1) When n is divided by 15, the quotient is 3k and the remainder is 5j 
(2) When n is divided by 9, the quotient is 5k and the remainder is 5j

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gracie · Answer

n and k are positive integers. When n is divided by 23, the quotient is 2k and the remainder is j. What is the value of j + k?

(1) When n is divided by 15, the quotient is 3k and the remainder is 5j
(2) When n is divided by 9, the quotient is 5k and the remainder is 5j

n/23=2k+j
equation 1: n=46k+j
n/15=3k+5j
equation 2: n=45k+5j
subtracting e1 from e2, 
k=4j
insufficient
n/9=5k+5j
remainder 5j must be less than divisor 9
if j is less than 2, then j=1
k=4j=4
j+k=5
sufficient

BrentGMATPrepNow · Answer

There are two important rules regarding remainders:

Rule #1:  "  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

Rule #2:    When positive integer N is divided by positive integer D, the remainder R is such that 0  <  R < D  
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0

---------------------------

Target question :    What is the value of j + k?

Given : When n is divided by 23, the quotient is 2k and the remainder is j  
From Rule #1, we can write: n = (23)(2k) + j
Simplify:  n = 46k + j

Statement 1 : When n is divided by 15, the quotient is 3k and the remainder is 5j   
From rule #1, we can write: n = (15)(3k) + 5j
Simplify: n = 45k + 5j
Since we also know that  n = 46k + j , we can write:  46k + j  = 45k + 5j
Simplify to get: k = 4j

NOTE: From rule #2, we know that the remainder must be LESS THAN 15
Since we're told that j is a positive integer, this means 5j can equal either 5 (if j = 1) or 10 (if j = 2). 
This means there are two possible cases: 
case a: j = 1: Since k = 4j, this tells us that k = 4, which means  j + k = 1 + 4 = 5 
case b: j = 2: Since k = 4j, this tells us that k = 8, which means  j + k = 2 + 8 = 10 
Since we cannot answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : When n is divided by 9, the quotient is 5k and the remainder is 5j  
From rule #1, we can write: n = (9)(5k) + 5j
Simplify: n = 45k + 5j
Since we also know that  n = 46k + j , we can write:  46k + j  = 45k + 5j
Simplify to get: k = 4j

NOTE: From rule #2, we know that the remainder must be LESS THAN 9
Since we're told that j is a positive integer, this means 5j MUST equal either 5 (when j = 1)
So, we KNOW that j = 1
We also know that k = 4j, which means k = 4
So, we can be CERTAIN that   j + k = 1 + 4 = 5 
Since we can answer the  target question  with certainty, statement 2 is SUFFICIENT

Answer:  B

Cheers,
Brent

SVaidyaraman · Answer

Given quotient when divided by 23 is 2k and the remainder is j

We will have values 1,2,3,4 etc for k and find out.

Statement 1: We see at k=4, the numbers can range from 184 to 194. Remainder j and 5j means, the number is 185 and j=1 . But, when k =8, the numbers range from 368 to 374. Remainder j and 5j means the number is 370 and j =2. So info not sufficient to determine j and k exactly as there are 2 possible values of j and k.

Statement 2: when k=4, the numbers can range from 184 to 189. Remainder j and 5j means the number is 185.

In the second case we will have only one range of numbers that accommodate both j and 5j. Hence statement 2 is sufficient.

So what is tested is one's understanding that the remainder cannot be greater than 9 in the second case i.e., j being 2 which was possible in the case of 15.

ledzep10 · Answer

We are told that n and k are positive integers, but it isn't mentioned that j needs to be a positive integer. In the second statement we get j<9/5. We have k = 4j and k has to be a positive integer. Therefore it satisfies if j = 1, but doesn't it also satisfy for j = 1.25? Because 4 x 1.25 gives an integer for k. So we can get j+k = 5 and 6.25. It is not mentioned that j+k have to be an integer either.

Bunuel · Answer

j is a remainder. It must be a non-negative integer by definition.

P.S. Pure algebraic questions are no longer a part of the  DS syllabus  of the GMAT.

DS questions in GMAT Focus encompass various types of word problems, such as:

Word Problems 
 Work Problems 
 Distance Problems 
 Mixture Problems 
 Percent and Interest Problems 
 Overlapping Sets Problems 
 Statistics Problems 
 Combination and Probability Problems

While these questions may involve or necessitate knowledge of algebra, arithmetic, inequalities, etc., they will always be presented in the form of word problems. You won’t encounter pure "algebra" questions like, "Is x > y?" or "A positive integer n has two prime factors..."

Check  GMAT Syllabus for Focus Edition

You can also visit the  Data Sufficiency forum  and filter questions by  OG 2024-2025, GMAT Prep (Focus), and Data Insights Review 2024-2025 sources  to see the types of questions currently tested on the GMAT.

So, you can ignore this question.

Hope it helps.

n and k are positive integers. When n is divided by 23, the quotient i

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