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

It is currently 23 Jul 2018, 08:53

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

Finding the remainder when dividing negative numbers

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

Hide Tags

Manager
Manager
avatar
Joined: 27 Apr 2008
Posts: 180
Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 12:36
Suppose I want to divide -11 by 5. What is the the remainder and quotient?
Intern
Intern
User avatar
Joined: 22 Dec 2009
Posts: 13
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 13:26
Just like in multiplication, when one but not both numbers are negative, the answer will be negative. If both numbers are negative, the answer will be positive. In number theory, remainders are always positive. -11 divided by 5 equals -2 remainder 1.
Manager
Manager
avatar
Joined: 27 Apr 2008
Posts: 180
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 14:09
handsomebrute wrote:
Just like in multiplication, when one but not both numbers are negative, the answer will be negative. If both numbers are negative, the answer will be positive. In number theory, remainders are always positive. -11 divided by 5 equals -2 remainder 1.


But according to the formula y=xq + r (q = quotient, r = remainder):

-11 = 5q + r

If q = -2 and r = 1 as you said, then:

-11 = 5(-2) + (1)
-11 = -10 + 1
-11 = -9

Did I miss a step somewhere?
Senior Manager
Senior Manager
User avatar
Joined: 22 Dec 2009
Posts: 320
GMAT ToolKit User
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 14:09
handsomebrute wrote:
Just like in multiplication, when one but not both numbers are negative, the answer will be negative. If both numbers are negative, the answer will be positive. In number theory, remainders are always positive. -11 divided by 5 equals -2 remainder 1.


I doubt this...

- 11 = 5 x (-2) + 1 ????

Guess remainder should be -1. It would then satisfy the equation above!

Cheers!
JT
_________________

Cheers!
JT...........
If u like my post..... payback in Kudos!! :beer

|Do not post questions with OA|Please underline your SC questions while posting|Try posting the explanation along with your answer choice|
|For CR refer Powerscore CR Bible|For SC refer Manhattan SC Guide|


~~Better Burn Out... Than Fade Away~~

Manager
Manager
avatar
Joined: 27 Apr 2008
Posts: 180
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 14:25
jeeteshsingh wrote:
I doubt this...

- 11 = 5 x (-2) + 1 ????

Guess remainder should be -1. It would then satisfy the equation above!

Cheers!
JT


Can you have negative remainders? Never heard of such a thing before...
Senior Manager
Senior Manager
User avatar
Joined: 22 Dec 2009
Posts: 320
GMAT ToolKit User
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 14:33
mrblack wrote:
jeeteshsingh wrote:
I doubt this...

- 11 = 5 x (-2) + 1 ????

Guess remainder should be -1. It would then satisfy the equation above!

Cheers!
JT


Can you have negative remainders? Never heard of such a thing before...


No offence mate.. but I havent heard of anything as positive remainder :lol: (on a lighter note)

Usually we simply call them as remainders as long as I remember...
_________________

Cheers!
JT...........
If u like my post..... payback in Kudos!! :beer

|Do not post questions with OA|Please underline your SC questions while posting|Try posting the explanation along with your answer choice|
|For CR refer Powerscore CR Bible|For SC refer Manhattan SC Guide|


~~Better Burn Out... Than Fade Away~~

2 KUDOS received
Senior Manager
Senior Manager
User avatar
Joined: 22 Dec 2009
Posts: 320
GMAT ToolKit User
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 14:58
2
OK... finally I got some insight on this..

There is way of calculating the remainder when dealing with Negative Numbers....

Say you have a Number 'N' which is to be divided by 'd'. On division you get a quotient 'q' and remainder 'r'

Now if N is +ve, then
\(N = q*d + r\)
where \(q*d \leq N \leq (q+1)*d\) and \(r = N - q*d\)

e.g. \(11 = 5*2 + 1\)

Now if N is negative, we have a change.... in this equation:
\(q*d \leq N \leq (q+1)*d\)... mutliply by -ve throughout and see the change as:
\(-(q+1)*d \leq -N \leq -(q)*d\)

Therefore remainder now would be:
\(r = (-N) - (-(q+1)*d) = (q+1)*d -N\)

e.g. Let N = -11, d = 5...
Hence \(-(3)*5 \leq -11 \leq -(2)*5\)
Therefore \(q = 2\)
Hence \(r = (2+1)*5-11 = 4\)

You can verify this as \(-11 = 5 * (-3) + 4\)
Therefore quotient = -3 and remainder = 4!
Hope it makes sence and is clear....

Cheers!
JT
_________________

Cheers!
JT...........
If u like my post..... payback in Kudos!! :beer

|Do not post questions with OA|Please underline your SC questions while posting|Try posting the explanation along with your answer choice|
|For CR refer Powerscore CR Bible|For SC refer Manhattan SC Guide|


~~Better Burn Out... Than Fade Away~~

Manager
Manager
avatar
Joined: 27 Aug 2009
Posts: 100
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 15:16
Reminder can be either positive or negative
Example:
If a and d are integers, with d non-zero, then a remainder is an integer r such that a = qd + r for some integer q, and with 0 ≤ |r| < |d|.
When defined this way, there are two possible remainders. For example, the division of −42 by −5 can be expressed as either
−42 = 9×(−5) + 3
as is usual for mathematicians, or
−42 = 8×(−5) + (−2).
So the remainder is then either 3 or −2.

the negative remainder is obtained from the positive one just by subtracting 5, which is d. This holds in general. When dividing by d, if the positive remainder is r1, and the negative one is r2, then
r1 = r2 + d.
1 KUDOS received
Senior Manager
Senior Manager
User avatar
Joined: 22 Dec 2009
Posts: 320
GMAT ToolKit User
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 15:51
1
dmetla wrote:
Reminder can be either positive or negative
Example:
If a and d are integers, with d non-zero, then a remainder is an integer r such that a = qd + r for some integer q, and with 0 ≤ |r| < |d|.
When defined this way, there are two possible remainders. For example, the division of −42 by −5 can be expressed as either
−42 = 9×(−5) + 3
as is usual for mathematicians, or
−42 = 8×(−5) + (−2).
So the remainder is then either 3 or −2.

the negative remainder is obtained from the positive one just by subtracting 5, which is d. This holds in general. When dividing by d, if the positive remainder is r1, and the negative one is r2, then
r1 = r2 + d.


I think this is not correct because.....

in −42 = 8×(−5) + (−2)... you are dividing -42 with 8 and choosing -5 as quotient...but -5 x 8 gives -40 which is greater than -42 and hence is not as per the divisibility methodology....

If u have to divide 11 by 5, you would use 10 (5x2) and not 15 (5x3) as 10 is less 11..
Similarly, for -42 you should use -48 (i.e. 8x-6) as -48 is less than -42 but -40 is more than -42!

The eq −42 = 8×(−5) + (−2) is correct mathematically but fails to ascertain the quotient and remainder correctly...!

More over your first example (highlighted in red) is wrong.. as you are dividing -42 with -5 which means you dividing 42 with 5 and hence the situation of a negative number being divided is void!
_________________

Cheers!
JT...........
If u like my post..... payback in Kudos!! :beer

|Do not post questions with OA|Please underline your SC questions while posting|Try posting the explanation along with your answer choice|
|For CR refer Powerscore CR Bible|For SC refer Manhattan SC Guide|


~~Better Burn Out... Than Fade Away~~

Expert Post
3 KUDOS received
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 47221
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 17:09
3
5
Wiki definition of the remainder:
If \(a\) and \(d\) are natural numbers, with \(d\) non-zero, it can be proven that there exist unique integers \(q\) and \(r\), such that \(a = qd + r\) and \(0\leq{r} < d\). The number \(q\) is called the quotient, while \(r\) is called the remainder.


GMAT Prep definition of the remainder:
If \(a\) and \(d\) are positive integers, there exists unique integers \(q\) and \(r\), such that \(a = qd + r\) and \(0\leq{r}<d\). \(q\) is called a quotient and \(r\) is called a remainder.

Moreover many GMAT books say factor is a "positive divisor", \(d>0\).

I've never seen GMAT question asking the ramainder when dividend (\(a\)) is negative, but if we'll cancel this restriction (\(dividend=a<0\)), and only this restriction, meaning that we'll leave the other one (\(0\leq{r}<d\)), then division of \(-11\) by \(5\) will result:

\(0\leq{r}<d\), \(a=qd + r\) --> \(0\leq{r}<5\), \(-11=(-3)*5+4\). Hence \(remainder=r=4\).

Hope it helps.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Manager
Manager
avatar
Joined: 27 Apr 2008
Posts: 180
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 06 Jan 2010, 19:51
Bunuel wrote:
Wiki definition of the remainder:
If \(a\) and \(d\) are natural numbers, with \(d\) non-zero, it can be proven that there exist unique integers \(q\) and \(r\), such that \(a = qd + r\) and \(0\leq{r} < d\). The number \(q\) is called the quotient, while \(r\) is called the remainder.


GMAT Prep definition of the remainder:
If \(a\) and \(d\) are positive integers, there exists unique integers \(q\) and \(r\), such that \(a = qd + r\) and \(0\leq{r}<d\). \(q\) is called a quotient and \(r\) is called a remainder.

Moreover many GMAT books say factor is a "positive divisor", \(d>0\).

I've never seen GMAT question asking the ramainder when dividend (\(a\)) is negative, but if we'll cancel this restriction (\(dividend=a<0\)), and only this restriction, meaning that we'll leave the other one (\(0\leq{r}<d\)), then division of \(-11\) by \(5\) will result:

\(0\leq{r}<d\), \(a=qd + r\) --> \(0\leq{r}<5\), \(-11=(-3)*5+4\). Hence \(remainder=r=4\).

Hope it helps.


Thanks for the lengthy response Bunuel. The wiki entry seems to imply that there could be a +ve or -ve remainder. Referring to my original question once more, it seems that -11/5 can be either:

-11 = (-3)*5 + 4
or
-11 = (-2)*5 + (-1)

I think either case is valid.
Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 47221
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 09 Jan 2010, 00:53
mrblack wrote:
Bunuel wrote:
Wiki definition of the remainder:
If \(a\) and \(d\) are natural numbers, with \(d\) non-zero, it can be proven that there exist unique integers \(q\) and \(r\), such that \(a = qd + r\) and \(0\leq{r} < d\). The number \(q\) is called the quotient, while \(r\) is called the remainder.


GMAT Prep definition of the remainder:
If \(a\) and \(d\) are positive integers, there exists unique integers \(q\) and \(r\), such that \(a = qd + r\) and \(0\leq{r}<d\). \(q\) is called a quotient and \(r\) is called a remainder.

Moreover many GMAT books say factor is a "positive divisor", \(d>0\).

I've never seen GMAT question asking the ramainder when dividend (\(a\)) is negative, but if we'll cancel this restriction (\(dividend=a<0\)), and only this restriction, meaning that we'll leave the other one (\(0\leq{r}<d\)), then division of \(-11\) by \(5\) will result:

\(0\leq{r}<d\), \(a=qd + r\) --> \(0\leq{r}<5\), \(-11=(-3)*5+4\). Hence \(remainder=r=4\).

Hope it helps.


Thanks for the lengthy response Bunuel. The wiki entry seems to imply that there could be a +ve or -ve remainder. Referring to my original question once more, it seems that -11/5 can be either:

-11 = (-3)*5 + 4
or
-11 = (-2)*5 + (-1)

I think either case is valid.


Wiki is not a best source for Math but even its definition: \(0\leq{r} < d\), so \(0\leq{r}\), remainder must be positive. But again I wouldn't worry about this issue for GMAT.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Manager
Manager
User avatar
Joined: 01 Jan 2008
Posts: 220
Schools: Booth, Stern, Haas
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 07 Jul 2010, 08:35
Bunuel wrote:
Wiki definition of the remainder:
If \(a\) and \(d\) are natural numbers, with \(d\) non-zero, it can be proven that there exist unique integers \(q\) and \(r\), such that \(a = qd + r\) and \(0\leq{r} < d\). The number \(q\) is called the quotient, while \(r\) is called the remainder.


GMAT Prep definition of the remainder:
If \(a\) and \(d\) are positive integers, there exists unique integers \(q\) and \(r\), such that \(a = qd + r\) and \(0\leq{r}<d\). \(q\) is called a quotient and \(r\) is called a remainder.

Moreover many GMAT books say factor is a "positive divisor", \(d>0\).

I've never seen GMAT question asking the ramainder when dividend (\(a\)) is negative, but if we'll cancel this restriction (\(dividend=a<0\)), and only this restriction, meaning that we'll leave the other one (\(0\leq{r}<d\)), then division of \(-11\) by \(5\) will result:

\(0\leq{r}<d\), \(a=qd + r\) --> \(0\leq{r}<5\), \(-11=(-3)*5+4\). Hence \(remainder=r=4\).

Hope it helps.


Hi, bunuel

how did you come up with -3, I mean why exactly -3 and not -2 or -17?
Manager
Manager
avatar
Joined: 06 Jun 2014
Posts: 51
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 02 May 2015, 11:57
This is what I think and according to the reminder theory :

when a=b*c => reminder of a/x i same as reminder of (b*c)/x
moreover it same as reminder of (reminder of b/x * reminder ofc/x)/x

I will try to ilsitrate with the example. actualy this concept I saw on the forum somewhere posted by EvaJager I think.

-11 = -1 *11 => R of -1*11/5 is Rof (R of-1/5 * R of 11/5)/5

now R of -1/5 is -1 and R of 11/5 is 1 => R of -1*1/5 is just R of-1/5 which is -1. since we cant really have negative reminder we need to add the negative

reminder back to the divisor 5, or 5-1 = 4, so 4 is the reminder
Expert Post
Math Revolution GMAT Instructor
User avatar
V
Joined: 16 Aug 2015
Posts: 5878
GMAT 1: 760 Q51 V42
GPA: 3.82
Premium Member
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 21 Sep 2016, 20:12
As always, questions like CMT 3,4 are often on the exam in a more developed form. Take a look at below and you’ll see the type of question that’s really common these days. This one, too, is a 5051-evel problem that is equivalent to CMT 4(A). You must get used to these. You need to know how variable approaches and CMT are related.

(ex 1) (integer) If x and y are postive integers, what is the remainder when 100x+y is divided by 11?
1) x=22
2) y=1
==> If you change the original condition and the prolem, you all ways get the remainder of 1 if you divide 100x by 11 regardless of the value of x. Thus, you only need to know y. Therefore, the answer is . This is a typical 5051 level problem. (CMT 4(A))
Answer: A
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $99 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

Manager
Manager
avatar
Joined: 28 Jun 2016
Posts: 207
Location: Canada
Concentration: Operations, Entrepreneurship
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 21 Sep 2016, 21:14
MathRevolution wrote:
As always, questions like CMT 3,4 are often on the exam in a more developed form. Take a look at below and you’ll see the type of question that’s really common these days. This one, too, is a 5051-evel problem that is equivalent to CMT 4(A). You must get used to these. You need to know how variable approaches and CMT are related.

(ex 1) (integer) If x and y are postive integers, what is the remainder when 100x+y is divided by 11?
1) x=22
2) y=1
==> If you change the original condition and the prolem, you all ways get the remainder of 1 if you divide 100x by 11 regardless of the value of x. Thus, you only need to know y. Therefore, the answer is . This is a typical 5051 level problem. (CMT 4(A))
Answer: A



Is it B or A? I am getting C.

St 1:

y can be anything

St 2:

For 101---- r=2
For 201 ----r=3
Intern
Intern
User avatar
Joined: 11 Nov 2016
Posts: 4
Re: Finding the remainder when dividing negative numbers  [#permalink]

Show Tags

New post 24 Sep 2017, 09:29
handsomebrute wrote:
Just like in multiplication, when one but not both numbers are negative, the answer will be negative. If both numbers are negative, the answer will be positive. In number theory, remainders are always positive. -11 divided by 5 equals -2 remainder 1.


Can the quotient be negative?

as per OG

If x and y are positive integers, there exist unique integers q and r , called the quotient and remainder, respectively, such that y = xq + r and 0 ≤ r < x . For example, when 28 is divided by 8, the quotient is 3 and the remainder is 4 since 28 = (8)(3) + 4. Note that y is divisible by x if and only if the remainder r is 0; for example, 32 has a remainder of 0 when divided by 8 because 32 is divisible by 8. Also, note that when a smaller integer is divided by a larger integer, the quotient is 0 and the remainder is the smaller integer. For example, 5 divided by 7 has the quotient 0 and the remainder 5 since 5 = (7)(0) + 5.

So quotient can't be negative?
Re: Finding the remainder when dividing negative numbers &nbs [#permalink] 24 Sep 2017, 09:29
Display posts from previous: Sort by

Finding the remainder when dividing negative numbers

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

Events & Promotions

PREV
NEXT


GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

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

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.