GMAT Question of the Day - Daily to your Mailbox; hard ones only

It is currently 20 Aug 2019, 15:51

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

If m and n are positive integers such that m > n, what is the Re.

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Find Similar Topics 
Current Student
User avatar
D
Joined: 12 Aug 2015
Posts: 2604
Schools: Boston U '20 (M)
GRE 1: Q169 V154
GMAT ToolKit User
If m and n are positive integers such that m > n, what is the Re.  [#permalink]

Show Tags

New post 17 Oct 2016, 13:14
1
2
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

77% (01:32) correct 23% (01:39) wrong based on 143 sessions

HideShow timer Statistics

If m and n are positive integers such that m > n, what is the remainder when m^2 – n^2 is divided by 21?

Statement 1: The remainder when (m + n) is divided by 21 is 1.

Statement 2: The remainder when (m – n) is divided by 21 is 1.

_________________
Board of Directors
User avatar
D
Status: QA & VA Forum Moderator
Joined: 11 Jun 2011
Posts: 4573
Location: India
GPA: 3.5
WE: Business Development (Commercial Banking)
GMAT ToolKit User
Re: If m and n are positive integers such that m > n, what is the Re.  [#permalink]

Show Tags

New post 17 Oct 2016, 13:30
stonecold wrote:
If m and n are positive integers such that m > n, what is the remainder when m^2 – n^2 is divided by 21?

Statement 1: The remainder when (m + n) is divided by 21 is 1.

Statement 2: The remainder when (m – n) is divided by 21 is 1.


FROM STATEMENT - I ( INSUFFICIENT)

We do not have any relationship between the value of m & n as such it won't be possible for us to find the remainder of \(\frac{(m^2 – n^2)}{21}\), since m & n can take any values...

FROM STATEMENT - I ( INSUFFICIENT)

We do not have any relationship between the value of m & n as such it won't be possible for us to find the remainder of \(\frac{(m^2 – n^2)}{21}\), since m & n can take any values...

COMBINE STATEMENT I & II (SUFFICIENT)

\(\frac{(m^2 – n^2)}{21} = \frac{( m+n )( m-n )}{21}\) = \(Remainder \ 1\)

Hence BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked, answer will be (C)


Similar Question to practice Here
_________________
Thanks and Regards

Abhishek....

PLEASE FOLLOW THE RULES FOR POSTING IN QA AND VA FORUM AND USE SEARCH FUNCTION BEFORE POSTING NEW QUESTIONS

How to use Search Function in GMAT Club | Rules for Posting in QA forum | Writing Mathematical Formulas |Rules for Posting in VA forum | Request Expert's Reply ( VA Forum Only )
SVP
SVP
User avatar
V
Status: It's near - I can see.
Joined: 13 Apr 2013
Posts: 1689
Location: India
Concentration: International Business, Operations
Schools: INSEAD Jan '19
GPA: 3.01
WE: Engineering (Real Estate)
Reviews Badge
Re: If m and n are positive integers such that m > n, what is the Re.  [#permalink]

Show Tags

New post 29 Mar 2018, 02:34
stonecold wrote:
If m and n are positive integers such that m > n, what is the remainder when m^2 – n^2 is divided by 21?

Statement 1: The remainder when (m + n) is divided by 21 is 1.

Statement 2: The remainder when (m – n) is divided by 21 is 1.


Bunuel, pushpitkc : Please bring some light on this question using number plugging technique.

QZ
_________________
"Do not watch clock; Do what it does. KEEP GOING."
CEO
CEO
User avatar
V
Joined: 12 Sep 2015
Posts: 3912
Location: Canada
Re: If m and n are positive integers such that m > n, what is the Re.  [#permalink]

Show Tags

New post 02 Jan 2019, 12:11
Top Contributor
stonecold wrote:
If m and n are positive integers such that m > n, what is the remainder when m² – n² is divided by 21?

(1) The remainder when (m + n) is divided by 21 is 1.
(2) The remainder when (m – n) is divided by 21 is 1.

Given: m and n are positive integers such that m > n

Target question: What is the remainder when m² – n² is divided by 21?

Statement 1: The remainder when (m + n) is divided by 21 is 1
There are several values of m and n that satisfy statement 1. Here are two:
Case a: m = 12 and n = 10. This means m + n = 12 + 10 = 22, and 22 divided by 21 leaves remainder 1. In this case, m² – n² = 12² – 10² = 144 - 100 = 44.
When we divide 44 by 21, we get 2 with remainder 2. So, the answer to the target question is when m² – n² is divided by 21, the remainder is 2
Case b: m = 13 and n = 9. This means m + n = 13 + 9 = 22, and 22 divided by 21 leaves remainder 1. In this case, m² – n² = 13² – 9² = 169 - 81 = 88.
When we divide 88 by 21, we get 4 with remainder 4. So, the answer to the target question is when m² – n² is divided by 21, the remainder is 4
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The remainder when (m – n) is divided by 21 is 1
There are several values of m and n that satisfy statement 2. Here are two:
Case a: m = 5 and n = 4. This means m - n = 5 - 4 = 1, and 1 divided by 21 leaves remainder 1. In this case, m² – n² = 5² – 4² = 25 - 16 = 9.
When we divide 9 by 21, we get 0 with remainder 9. So, the answer to the target question is when m² – n² is divided by 21, the remainder is 9
Case b: m = 4 and n = 3. This means m - n = 4 - 3 = 1, and 1 divided by 21 leaves remainder 1. In this case, m² – n² = 4² – 3² = 16 - 9 = 7.
When we divide 7 by 21, we get 0 with remainder 7. So, the answer to the target question is when m² – n² is divided by 21, the remainder is 7
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that the remainder is 1 when (m + n) is divided by 21
In other words, m+n is 1 greater than some multiple of 21.
So, we can write: m+n = 21k + 1 (for some integer k)

Statement 2 tells us that the remainder is 1 when (m - n) is divided by 21
In other words, m-n is 1 greater than some multiple of 21.
So, we can write: m-n = 21j + 1 (for some integer j)

Now recognize that we can factor m² – n²
We get: m² – n² = (m + n)(m - n)
= (21k + 1)(21j + 1)
= 21²mn + 21k + 21j + 1
= 21(21mn + k + j) + 1
Since 21(21mn + k + j) is definitely a multiple of 21, we can conclude that 21(21mn + k + j) + 1 is 1 greater than some multiple of 21.
In other words, m² – n² is 1 greater than some multiple of 21.
So, when m² – n² is divided by 21, the remainder is 1
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

RELATED VIDEO FROM OUR COURSE

_________________
Test confidently with gmatprepnow.com
Image
GMATH Teacher
User avatar
P
Status: GMATH founder
Joined: 12 Oct 2010
Posts: 937
Re: If m and n are positive integers such that m > n, what is the Re.  [#permalink]

Show Tags

New post 03 Jan 2019, 05:13
stonecold wrote:
If m and n are positive integers such that m > n, what is the remainder when m^2 – n^2 is divided by 21?

Statement 1: The remainder when (m + n) is divided by 21 is 1.

Statement 2: The remainder when (m – n) is divided by 21 is 1.

Important: the solution presented below assumes a reasonable quantitative maturity.
(If you do not have that, I recommend the excellent step-by-step presentation offered by Brent above.)


\(m > n \ge 1\,\,\,{\rm{ints}}\,\,\,\left( * \right)\)

\({m^{\rm{2}}} - {n^2} = 21K + R\)

\(K,R\,\,{\rm{ints}}\,\,,\,\,\,0 \le R \le 20\)

\(? = R\)


\(\left( 1 \right)\,\,m + n = 21J + 1\,\,,\,\,\,J\mathop \ge \limits^{\left( * \right)} 1\,\,\,{\mathop{\rm int}}\)

\(\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {21,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{m^{\rm{2}}} - {n^2} = {21^2} - {1^2}\,\,\, \Rightarrow \,\,\,\,R = \,\,20 \hfill \cr
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {20,2} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{m^{\rm{2}}} - {n^2} = {20^2} - {2^2} = \left( {20 - 2} \right)\left( {20 + 2} \right) = \left( {21 - 3} \right)\left( {21 + 1} \right) = 21\left( {21 + 1 - 3} \right) - 3\,\,\, \Rightarrow \,\,\,\,R = \,\,18\, \hfill \cr} \right.\)


\(\left( 2 \right)\,\,m - n = 21L + 1\,\,,\,\,\,L\mathop \ge \limits^{\left( * \right)} 0\,\,\,{\mathop{\rm int}} \,\)

\(\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,1} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{m^{\rm{2}}} - {n^2} = 3\,\,\, \Rightarrow \,\,\,\,R = \,\,3 \hfill \cr
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {3,2} \right)\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{m^{\rm{2}}} - {n^2} = 5\,\,\, \Rightarrow \,\,\,\,R = \,5\, \hfill \cr} \right.\)


\(\left( {1 + 2} \right)\,\,\,\,\left( {21J + 1} \right)\left( {21L + 1} \right) = 21\left( {21JL + J + L} \right) + 1\,\,\, \Rightarrow \,\,\,\,R = \,\,1\,\,\,\,\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
GMAT Club Bot
Re: If m and n are positive integers such that m > n, what is the Re.   [#permalink] 03 Jan 2019, 05:13
Display posts from previous: Sort by

If m and n are positive integers such that m > n, what is the Re.

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  





Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne