Last visit was: 19 Nov 2025, 16:17 It is currently 19 Nov 2025, 16:17
Home
GMAT
Messages
Tests
Schools
Events
Rewards
Chat
My Profile
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.
Go to My Error Log Learn more
Close
Request Expert Reply
Confirm Cancel
Back to Forum
Create Topic
   1   2 
User avatar
jfranciscocuencag
User avatar
Joined: 12 Sep 2017
Last visit: 17 Aug 2024
Posts: 227
Own Kudos:
140
 [1]
Given Kudos: 132
Posts: 227
Kudos: 140
 [1]
Compilation of tips and tricks to deal with remainders. 05 Jan 2019, 14:01
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
sriharimurthy
Hi guys,
This is in conjunction with another post which has questions dealing with remainders (https://gmatclub.com/forum/collection-of ... 74776.html). I'm just trying to put together a list of tips and tricks which we can use to solve these kind of problems with greater accuracy and speed. Please feel free to comment and make suggestions. Hopefully we can add onto this list and cover all sorts of strategies that would help us deal with remainders!
Cheers.

Please read this first :
1) Take your time with these points. Some of them might be a little difficult to follow in the first reading, but don't give up. The concepts are fairly simple.
2) These tips if mastered will be extremely valuable in the GMAT to help solve a variety of questions not limited specifically to remainders. I have been using them for quite a while now and they have not only helped me improve my accuracy but also my speed.
3) If you have any doubts, please do not hesitate to ask (no matter how stupid you might think them to be!). If you do not ask, you will never learn.
4) Lastly, have fun while trying to understand these tips and tricks as that, according to me, is the best possible way to learn.

All the best!

-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-
NOTE: Where ever you see R of 'x' it just stands for Remainder of x.


1) The possible remainders when a number is divided by ‘n’ can range from 0 to (n-1).
Eg. If n=10, possible remainders are 0,1,2,3,4,5,6,7,8 and 9.


2) If a number is divided by 10, its remainder is the last digit of that number. If it is divided by 100 then the remainder is the last two digits and so on.
This is good for questions such as : ' What is the last digit of.....' or ' What are the last two digits of.....' .


3) If a number leaves a remainder ‘r’ (the number is the divisor), all its factors will have the same remainder ‘r’ provided the value of ‘r’ is less than the value of the factor.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 7 (which is a factor of 21) will also be 5.

If the value of ‘r’ is greater than the value of the factor, then we have to take the remainder of ‘r’ divided by the factor to get the remainder.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 3 (which is a factor of 21) will be remainder of 5/3, which is 2.


4) Cycle of powers : This is used to find the remainder of \(n^x\), when divided by 10, as it helps us in figuring out the last digit of \(n^x\).

The cycle of powers for numbers from 2 to 10 is given below:

2: 2, 4, 8, 6 → all \(2^{4x}\) will have the same last digit.

3: 3, 9, 7, 1 → all \(3^{4x}\) will have the same last digit.

4: 4, 6 → all \(4^{2x}\) will have the same last digit.

5: 5 → all \(5^x\) will have the same last digit.

6: 6 → all \(6^x\) will have the same last digit.

7: 7, 9, 3, 1 → all \(7^{4x}\) will have the same last digit.

8: 8, 4, 2, 6 → all \(8^{4x}\) will have the same last digit.

9: 9, 1 → all \(9^{2x}\) will have the same last digit.

10: 0 → all \(10^x\) will have the same last digit.


5) Many seemingly difficult remainder problems can be simplified using the following formula :
\(R of \frac{x*y}{n} = R of \frac{(R of \frac{x}{n})*(R of \frac{y}{n})}{n}\)

Eg. \(R of \frac{20*27}{25} = R of \frac{(R of \frac{20}{25})*(R of \frac{27}{25})}{25} = R of \frac{(20)*(2)}{25} = R of \frac{40}{25} = 15\)


Eg. \(R of \frac{225}{13} = R of \frac{(15)*(15)}{13} = R of {(2)*(2)}{13} = R of \frac{4}{13} = 4\)


6) \(R of \frac{x*y}{n}\) , can sometimes be easier calculated if we take it as \(R of \frac{(R of \frac{(x-n)}{n})*(R of \frac{(y-n)}{n})}{n}\)
Especially when x and y are both just slightly less than n. This can be easier understood with an example:

Eg. \(R of \frac{(19)*(21)}{25} = R of \frac{(-6)*(-4)}{25} = 24\)

NOTE: Incase the answer comes negative, (if x is less than n but y is greater than n) then we have to simply add the remainder to n.

Eg. \(R of \frac{(23)*(27)}{25} = R of \frac{(-2)*(2)}{25} = -4.\) Now, since it is negative, we have to add it to 25.\(R = 25 + (-4) = 21\)


[Note: Go here to practice two good problems where you can use some of these concepts explained : https://gmatclub.com/forum/numbers-86325.html]


7) If you take the decimal portion of the resulting number when you divide by "n", and multiply it to "n", you will get the remainder. [Special thanks to h2polo for this one]

Note: Converse is also true. If you take the remainder of a number when divided by 'n', and divide it by 'n', it will give us the remainder in decimal format.

Eg. \(\frac{8}{5} = 1.6\)

In this case, \(0.6 * 5 = 3\)

Therefore, the remainder is \(3\).

This is important to understand for problems like the one below:

If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t?
(A) 2
(B) 4
(C) 8
(D) 20
(E) 45

OA :
Show Spoiler
E

THIS QUESTION IS DISCUSSED HERE.
Show more

Can someone please explain me the ROF formula?

Are x and y dividends or divisors?, Im confused :dazed
User avatar
swatjazz
User avatar
Joined: 22 Dec 2018
Last visit: 13 Jun 2022
Posts: 11
Own Kudos:
8
Given Kudos: 201
Concentration: Healthcare, International Business
WE:Medicine and Health (Healthcare/Pharmaceuticals)
Posts: 11
Kudos: 8
Compilation of tips and tricks to deal with remainders. 12 May 2019, 04:14
Kudos
Add Kudos
Bookmarks
Bookmark this Post
If a number leaves a remainder ‘r’ (the number is the divisor), all its factors will have the same remainder ‘r’ provided the value of ‘r’ is less than the value of the factor.
[If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 7 (which is a factor of 21) will also be 5.

I am having difficulty in understanding above. Can anyone please explain. Bunuel VeritasKarishma

Thank you!
avatar
SRH750
avatar
Joined: 01 Jun 2020
Last visit: 23 Sep 2021
Posts: 20
Own Kudos:
7
Given Kudos: 18
Posts: 20
Kudos: 7
Compilation of tips and tricks to deal with remainders. 12 Aug 2020, 22:55
Kudos
Add Kudos
Bookmarks
Bookmark this Post
This is a brilliant post !
User avatar
nj23598
User avatar
Joined: 10 Jun 2023
Last visit: 12 May 2024
Posts: 42
Own Kudos:
21
Given Kudos: 14
Location: India
Concentration: Finance, General Management
Schools: IIM-A '24
GMAT 1: 650 Q45 V34
GPA: 4
Schools: IIM-A '24
GMAT 1: 650 Q45 V34
Posts: 42
Kudos: 21
Compilation of tips and tricks to deal with remainders. 06 Aug 2023, 20:51
Kudos
Add Kudos
Bookmarks
Bookmark this Post
sriharimurthy
Hi guys,
This is in conjunction with another post which has questions dealing with remainders (https://gmatclub.com/forum/collection-o ... 74776.html). I'm just trying to put together a list of tips and tricks which we can use to solve these kind of problems with greater accuracy and speed. Please feel free to comment and make suggestions. Hopefully we can add onto this list and cover all sorts of strategies that would help us deal with remainders!
Cheers.

Please read this first :
1) Take your time with these points. Some of them might be a little difficult to follow in the first reading, but don't give up. The concepts are fairly simple.
2) These tips if mastered will be extremely valuable in the GMAT to help solve a variety of questions not limited specifically to remainders. I have been using them for quite a while now and they have not only helped me improve my accuracy but also my speed.
3) If you have any doubts, please do not hesitate to ask (no matter how stupid you might think them to be!). If you do not ask, you will never learn.
4) Lastly, have fun while trying to understand these tips and tricks as that, according to me, is the best possible way to learn.

All the best!

-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-
NOTE: Where ever you see R of 'x' it just stands for Remainder of x.


1) The possible remainders when a number is divided by ‘n’ can range from 0 to (n-1).
Eg. If n=10, possible remainders are 0,1,2,3,4,5,6,7,8 and 9.


2) If a number is divided by 10, its remainder is the last digit of that number. If it is divided by 100 then the remainder is the last two digits and so on.
This is good for questions such as : ' What is the last digit of.....' or ' What are the last two digits of.....' .


3) If a number leaves a remainder ‘r’ (the number is the divisor), all its factors will have the same remainder ‘r’ provided the value of ‘r’ is less than the value of the factor.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 7 (which is a factor of 21) will also be 5.

If the value of ‘r’ is greater than the value of the factor, then we have to take the remainder of ‘r’ divided by the factor to get the remainder.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 3 (which is a factor of 21) will be remainder of 5/3, which is 2.


4) Cycle of powers : This is used to find the remainder of \(n^x\), when divided by 10, as it helps us in figuring out the last digit of \(n^x\).

The cycle of powers for numbers from 2 to 10 is given below:

2: 2, 4, 8, 6 → all \(2^{4x}\) will have the same last digit.

3: 3, 9, 7, 1 → all \(3^{4x}\) will have the same last digit.

4: 4, 6 → all \(4^{2x}\) will have the same last digit.

5: 5 → all \(5^x\) will have the same last digit.

6: 6 → all \(6^x\) will have the same last digit.

7: 7, 9, 3, 1 → all \(7^{4x}\) will have the same last digit.

8: 8, 4, 2, 6 → all \(8^{4x}\) will have the same last digit.

9: 9, 1 → all \(9^{2x}\) will have the same last digit.

10: 0 → all \(10^x\) will have the same last digit.


5) Many seemingly difficult remainder problems can be simplified using the following formula :
\(R of \frac{x*y}{n} = R of \frac{(R of \frac{x}{n})*(R of \frac{y}{n})}{n}\)

Eg. \(R of \frac{20*27}{25} = R of \frac{(R of \frac{20}{25})*(R of \frac{27}{25})}{25} = R of \frac{(20)*(2)}{25} = R of \frac{40}{25} = 15\)


Eg. \(R of \frac{225}{13} = R of \frac{(15)*(15)}{13} = R of {(2)*(2)}{13} = R of \frac{4}{13} = 4\)


6) \(R of \frac{x*y}{n}\) , can sometimes be easier calculated if we take it as \(R of \frac{(R of \frac{(x-n)}{n})*(R of \frac{(y-n)}{n})}{n}\)
Especially when x and y are both just slightly less than n. This can be easier understood with an example:

Eg. \(R of \frac{(19)*(21)}{25} = R of \frac{(-6)*(-4)}{25} = 24\)

NOTE: Incase the answer comes negative, (if x is less than n but y is greater than n) then we have to simply add the remainder to n.

Eg. \(R of \frac{(23)*(27)}{25} = R of \frac{(-2)*(2)}{25} = -4.\) Now, since it is negative, we have to add it to 25.\(R = 25 + (-4) = 21\)


[Note: Go here to practice two good problems where you can use some of these concepts explained : https://gmatclub.com/forum/numbers-86325.html]


7) If you take the decimal portion of the resulting number when you divide by "n", and multiply it to "n", you will get the remainder. [Special thanks to h2polo for this one]

Note: Converse is also true. If you take the remainder of a number when divided by 'n', and divide it by 'n', it will give us the remainder in decimal format.

Eg. \(\frac{8}{5} = 1.6\)

In this case, \(0.6 * 5 = 3\)

Therefore, the remainder is \(3\).

This is important to understand for problems like the one below:

If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t?
(A) 2
(B) 4
(C) 8
(D) 20
(E) 45

OA :
Show Spoiler
E

THIS QUESTION IS DISCUSSED HERE.
Show more





Let a mod b be the remainder when a is divided by b

Note that xy mod b = ((x mod b)*(y mod b)) mod b

11 mod 7 = 4
11^2 mod 7 = 4*4 mod 7 =2
11^3 mod 7 = 2*4 mod 7 =1
11^4 mod 7 = 1*4 mod 7 = 4
... and then the cycle will repeat

So for 11^97 the remainder will be 4
User avatar
Lavanyakalapatapu
User avatar
Joined: 25 Apr 2021
Last visit: 19 May 2025
Posts: 7
Own Kudos:
2
Given Kudos: 46
Posts: 7
Kudos: 2
Compilation of tips and tricks to deal with remainders. 19 Sep 2024, 09:58
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hi, Can someone please explain pt.5 and pt.6 What is RoF?? How do we solve the equations.
sriharimurthy
Hi guys,
This is in conjunction with another post which has questions dealing with remainders (https://gmatclub.com/forum/collection-o ... 74776.html). I'm just trying to put together a list of tips and tricks which we can use to solve these kind of problems with greater accuracy and speed. Please feel free to comment and make suggestions. Hopefully we can add onto this list and cover all sorts of strategies that would help us deal with remainders!
Cheers.

Please read this first :
1) Take your time with these points. Some of them might be a little difficult to follow in the first reading, but don't give up. The concepts are fairly simple.
2) These tips if mastered will be extremely valuable in the GMAT to help solve a variety of questions not limited specifically to remainders. I have been using them for quite a while now and they have not only helped me improve my accuracy but also my speed.
3) If you have any doubts, please do not hesitate to ask (no matter how stupid you might think them to be!). If you do not ask, you will never learn.
4) Lastly, have fun while trying to understand these tips and tricks as that, according to me, is the best possible way to learn.

All the best!

-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-x-
NOTE: Where ever you see R of 'x' it just stands for Remainder of x.


1) The possible remainders when a number is divided by ‘n’ can range from 0 to (n-1).
Eg. If n=10, possible remainders are 0,1,2,3,4,5,6,7,8 and 9.


2) If a number is divided by 10, its remainder is the last digit of that number. If it is divided by 100 then the remainder is the last two digits and so on.
This is good for questions such as : ' What is the last digit of.....' or ' What are the last two digits of.....' .


3) If a number leaves a remainder ‘r’ (the number is the divisor), all its factors will have the same remainder ‘r’ provided the value of ‘r’ is less than the value of the factor.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 7 (which is a factor of 21) will also be 5.

If the value of ‘r’ is greater than the value of the factor, then we have to take the remainder of ‘r’ divided by the factor to get the remainder.
Eg. If remainder of a number when divided by 21 is 5, then the remainder of that same number when divided by 3 (which is a factor of 21) will be remainder of 5/3, which is 2.


4) Cycle of powers : This is used to find the remainder of \(n^x\), when divided by 10, as it helps us in figuring out the last digit of \(n^x\).

The cycle of powers for numbers from 2 to 10 is given below:

2: 2, 4, 8, 6 → all \(2^{4x}\) will have the same last digit.

3: 3, 9, 7, 1 → all \(3^{4x}\) will have the same last digit.

4: 4, 6 → all \(4^{2x}\) will have the same last digit.

5: 5 → all \(5^x\) will have the same last digit.

6: 6 → all \(6^x\) will have the same last digit.

7: 7, 9, 3, 1 → all \(7^{4x}\) will have the same last digit.

8: 8, 4, 2, 6 → all \(8^{4x}\) will have the same last digit.

9: 9, 1 → all \(9^{2x}\) will have the same last digit.

10: 0 → all \(10^x\) will have the same last digit.


5) Many seemingly difficult remainder problems can be simplified using the following formula :
\(R of \frac{x*y}{n} = R of \frac{(R of \frac{x}{n})*(R of \frac{y}{n})}{n}\)

Eg. \(R of \frac{20*27}{25} = R of \frac{(R of \frac{20}{25})*(R of \frac{27}{25})}{25} = R of \frac{(20)*(2)}{25} = R of \frac{40}{25} = 15\)


Eg. \(R of \frac{225}{13} = R of \frac{(15)*(15)}{13} = R of {(2)*(2)}{13} = R of \frac{4}{13} = 4\)


6) \(R of \frac{x*y}{n}\) , can sometimes be easier calculated if we take it as \(R of \frac{(R of \frac{(x-n)}{n})*(R of \frac{(y-n)}{n})}{n}\)
Especially when x and y are both just slightly less than n. This can be easier understood with an example:

Eg. \(R of \frac{(19)*(21)}{25} = R of \frac{(-6)*(-4)}{25} = 24\)

NOTE: Incase the answer comes negative, (if x is less than n but y is greater than n) then we have to simply add the remainder to n.

Eg. \(R of \frac{(23)*(27)}{25} = R of \frac{(-2)*(2)}{25} = -4.\) Now, since it is negative, we have to add it to 25.\(R = 25 + (-4) = 21\)


[Note: Go here to practice two good problems where you can use some of these concepts explained : https://gmatclub.com/forum/numbers-86325.html]


7) If you take the decimal portion of the resulting number when you divide by "n", and multiply it to "n", you will get the remainder. [Special thanks to h2polo for this one]

Note: Converse is also true. If you take the remainder of a number when divided by 'n', and divide it by 'n', it will give us the remainder in decimal format.

Eg. \(\frac{8}{5} = 1.6\)

In this case, \(0.6 * 5 = 3\)

Therefore, the remainder is \(3\).

This is important to understand for problems like the one below:

If s and t are positive integer such that s/t=64.12, which of the following could be the remainder when s is divided by t?
(A) 2
(B) 4
(C) 8
(D) 20
(E) 45

OA :
Show Spoiler
E

THIS QUESTION IS DISCUSSED HERE.
Show more
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 38,589
Own Kudos:
1,079
Posts: 38,589
Kudos: 1,079
Compilation of tips and tricks to deal with remainders. 22 Sep 2025, 11:15
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
   1   2 
Moderator:
Bunuel
Math Expert
105390 posts