Compilation of tips and tricks to deal with remainders.

Question

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.

h2polo · Accepted Answer

Here is another important property about reminders that everyone should understand:

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.

For example, 8/5 = 1.6

.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

Bunuel · Answer

Check the links below:

Remainders 
     Theory   
 REMAINDERS 
 Remainders: Tips and hints 
 Divisibility and Remainders on the GMAT 
 Remainders on the GMAT 
 A Remainders Shortcut for the GMAT 
 All About Negative Remainders on the GMAT 
 A Tricky Question on Negative Remainders 
 Compilation of tips and tricks to deal with remainders 
 Cyclicity and the remainders on the GMAT 
 GMAT Question Patterns: Remainders 
     Questions   
 Units digits, exponents, remainders problems  
 DS Questions 
 PS Questions

ctrlaltdel · Answer

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

Ans: Using the technique: remainder = 0.12*t => the answer is multiple of 12. but none of the options match...did i miss something or is my understanding wrong :oops

sriharimurthy · Answer

Don't worry. This question is not that straightforward but understanding it will really help you for any other such question.

The answer is a multiple of 0.12.

However, If we consider it to be a multiple of 12, then we have to multiply each of the answers by 100 and then check.

Try working it out now.

You should get the answer to be (E). If you face any further difficulties then feel free to ask again!

Shelen · Answer

Could you, please, explain the solution? I didn't get it.

I found out that the remaider is 3; should we just pick up the answer to get multiple of 3?

sriharimurthy · Answer

Hi Shelen,

In the question, it is given that s/t = 64.12

Thus, we know that the remainder in decimal format will be 0.12
(Note : Do not make the mistake of considering it to be 12. It is 0.12)

Now, we know that the answer should be a multiple of 0.12 since 'the remainder of s/t' will be equal to 'the remainder of s/t in decimal format' multiplied by 't'.

That is, R of (s/t) = 0.12*t ----> which is a multiple of 0.12 for all the positive integer values that 't' can hold.

Now, in order to make the calculation simpler, we can multiply  both sides  of the equation by 100.

R*100 = 12*t --> t = (R*100)/12

Now, since it is given that 't' is a positive integer, (R*100) has to be perfectly divisible by 12.

Thus look through the answer choices to see which one satisfies this condition.

You will find that R = 45 will be the only one that satisfies it, since 4500 is perfectly divisible by 12.

Therefore answer is choice (E) which is 45.

brownybuddy · Answer

Wondering if it is a good way to solve it as follows:

.12 = 12/100 = 3/25 implies that the remainder is 3 or multiple of 3.

Comments please.

wilbase · Answer

I tested it with some numbers. (8/5, 9/4, 57/12, 57/15, 57/20) They are work using the way you described. I don't know how to prove it mathematically though. Does anyone know? This is a smart way if it works for all numbers!

sriharimurthy · Answer

Yes. It is supposed to be 25. Thanks for spotting that. I have edited it.

Yes. It follows from property number 7.

Since we are asked to find the remainder when 's' is divided by 't' and we are given the resulting number, we can write an equation as follows :

Remainder = (Decimal portion of the resulting number) * (Number we are dividing by)

Remainder = 0.12 * t

R =  \frac{12}{100}*t  =  \frac{3}{25}*t

So as you can see, the remainder 'R' must be a multiple of '3'  provided  't' is an integer.

Since we know that 't' is an integer, we can safely conclude that 'R' is a multiple of '3'.

Note :  In cases of remainder problems, even if 't' is not an integer it can be made into an integer. Eg. Remainder of 6/2.5 will be the same as Remainder of 12/5.

isiadeolumide33 · Answer

Hi ,

I am new on this forum , and i will tell you that what i have gained reading all the posts in the various sections have been mind blowing .

I wish to study indepth before i sit for my GMAT. I am quite ambitious with the kind of score i desire and i think you guys are the best in terms of detailing the requirements .

Please kindly explain the 3rd to 5th Rule on the remainder lecture ... i cant seem to grasp the rules!!

Thank you

Easy

msunny · Answer

Property 3 says that

But if we see 21/5 remainder is 1
7 is a factor of 21
7/5 and the remainder is 2.

Why the contradiction ??

sriharimurthy · Answer

Hi,

To make this example more clear :  If any number when divided by 21 leaves a remainder of 5, then that number when divided by any factor of 21 will also leave a remainder of 5 provided the remainder is less than the factor.
 
Eg. R of 26/21 = 5

Factors of 21 are 3 and 7

Since 7 is greater than 5, R of 26/7 = 5

Since 3 is less than 5, R of 26/3 = R of 5/3 = 2

Hope this makes it clear.

I think I will edit the main post to make this point less confusing.

Cheers.

sriharimurthy · Answer

Hi, 3rd Rule : I have explained the 3rd rule in the above post. 4th Rule : The cycle of powers is useful to know because it tells us the only possible values that the units place can hold for any particular number when it is raised to an integer power. Go through the following example to see the usefulness of this rule : 5th Rule : Again for this rule, the best way to understand it is to work through a couple of questions (numbers-86325.html). Go through my solutions for the two problems in the post I have linked and see how rules 5 and 6 relate to them. It might take a while for these concepts to get cemented but have a little patience and you will be rewarded. If you have any specific doubts you would like me to address then please let me know. Cheers.

Fokir · Answer

Hello,

Can some one please tell me how to solve the below?

what is the remainder of 11^97/ 7

How do we tackle these type of problems?

shrouded1 · Answer

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

russty · Answer

hi thanks a lot for these tips on remainders. Kudos to u.
I like the way u easily solved the 2 Remainder questions in some other posts. Could you also provide the links where you have solved questions involving the cycle of powers so as to get a better idea of that too.

thankssss

anuu · Answer

Hi, Here's a link regarding remainders and cycles of power. I found it very helpful. I guess this link was mentioned somewhere in this forum only.I had stored it in my favorite list..hope u too find it useful. https://takshzilabeta.com/cat-quant/numb ... rs-part-i/ Anu

gmatcrackerindia · Answer

An easy way to understand Number Tree:

kisna72 · Answer

great article. A suggestion - if you could add some examples of questions for each principle, it would be great for retention of the material. Thanks.

Compilation of tips and tricks to deal with remainders.

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards