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

 It is currently 06 Dec 2019, 08:20 ### 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

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.  # How To Solve: Remainder Problems

Author Message
TAGS:

### Hide Tags

GMAT Tutor G
Status: Tutor - BrushMyQuant
Joined: 05 Apr 2011
Posts: 622
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31 GPA: 3
WE: Information Technology (Computer Software)
How To Solve: Remainder Problems  [#permalink]

### Show Tags

12
29
How to Solve: Remainder Problems

Hi All,

I have learned a lot from gmatclub and am done with my gmat too. So, i have decided to contribute back.
As part of this i have decided to share the knowledge i have regarding various topics related to gmat quant.
Hope it will be useful. This post is about how to solve Remainder problems

Theory
Given that an integer "n" when divided by an integer "a" gives "r" as remainder then the integer "n" can be written as
n = ak + r
where k is a constant integer.

Example1. What is the remainder when B is divided by 6 if B is a positive integer?
(1) When B is divided by 18, the remainder is 3
(2) When B is divided by 12, the remainder is 9

Sol
STAT1 : When B is divided by 18, the remainder is 3
So, we can write B as
B = 18k + 3
Now, to check the remainder when B is divided by 6, we essentially need to check the remainder when 18k + 3 is divided by 6
18k goes with 6 so the remainder will 3
So, its sufficient

STAT2 : When B is divided by 12, the remainder is 9
So, we can write B as
B = 12k + 9
Now, to check the remainder when B is divided by 6, we essentially need to check the remainder when 12k + 9 is divided by 6
12k goes with 6 so the remainder will be the same as the remainder for 9 divided by 6 which is 3
So, remainder is 3
So, its sufficient

Example2:
What is the remainder when positive integer t is divided by 5?
(1) When t is divided by 4, the remainder is 1
(2) When t is divided by 3, the remainder is 1

Sol:
STAT1: When t is divided by 4, the remainder is 1
t = 4k +1
possible values of t are 1,5,9,13
Clearly we cannot find a unique remainder when t is divided by 5 as in some cases(t=1) we are getting the remainder as 1 and in some(t=5) we are getting the remainder as 0.
So, INSUFFICIENT

STAT2: When t is divided by 3, the remainder is 1
t = 3s + 1
possible values of t are 1,4,7,10,13,16,19
Clearly we cannot find a unique remainder when t is divided by 5 as in some cases(t=1) we are getting the remainder as 1 and in some(t=10) we are getting the remainder as 0.
So, INSUFFICIENT

taking both together
now there are two approaches
1. write the values of t from stat1 and then from stat2 and then take the common values
from STAT1 t = 1,5,9,13,17,21,25,29,33
from STAT2 t = 1,4,7,10,13,16,19,22,25,28,31,34
common values are t = 1,13,25,

2. equate t = 4k+1 to t=3s+1
we have 4k + 1 = 3s+1
k = 3s/4
since, k is an integer so only those values of s which are multiple of 4 will satisfy both STAT1 and STAT2
so, common values are given by t = 3s + 1 where s is multiple of 4
so t = 1,13,25 (for s=0,4,8 respectively)

Clearly we cannot find a unique remainder when t is divided by 5 as in some cases(t=1) we are getting the remainder as 1 and in some(t=10) we are getting the remainder as 0.
So, INSUFFICIENT

Example3:If p and n are positive integers and p > n, what is the remainder when p^2 - n^2 is divided by 15 ?
(1) The remainder when p + n is divided by 5 is 1.
(2) The remainder when p - n is divided by 3 is 1

Sol:
STAT1 : The remainder when p + n is divided by 5 is 1.
p+n = 5k + 1
but we cannot say anything about p^2 - n^2 just from this information.
So, INSUFFICIENT

STAT2 : The remainder when p - n is divided by 3 is 1
p-n = 3s + 1
but we cannot say anything about p^2 - n^2 just from this information.
So, INSUFFICIENT

Taking both together
p^2 - n^2 = (p+n) * (p-n) = (5k + 1) * (3s + 1)
= 15ks + 5k + 3s + 1
The remainder of the above expression by 15 is same as the remainder of 5k + 3s + 1 with 15 as 15ks will go with 15.
But we cannot say anything about the remainder as its value will change with the values of k and s.
So INSUFFICIENT

Example 4: If n is a positive integer and r is the remainder when 4 + 7n is divided by 3, what is the value of r?
(1) n+1 is divisible by 3
(2) n>20.

Sol:
r is the remainder when 4 + 7n is divided by 3
7n + 4 can we written as 6n + n + 3+ 1 = 3(2n+1) + n +1
remainder of 7n+4 by 3 will be same as remainder of 3(2n+1) + n +1 by 3
3*(2n+1) will go by 3 so the remainder will be the same as the remainder of (n+1) by 3

STAT1: n+1 is divisible by 3
n+1 = 3k (where k is an integer)
n+1 will give 0 as the remainder when divided by 3
so, 7n+4 will also give 0 as the remainder when its divided by 3 (as its remainder is same as the remainder for (n+1) when divided by 3)
=> r =0
So, SUFFICIENT

STAT2 n>20.
we cannot do anything by this information as there are many values of n
so, INSUFFICIENT.

Example 5: If x is an integer, is x between 27 and 54?
(1) The remainder when x is divided by 7 is 2.
(2) The remainder when x is divided by 3 is 2.

Sol:
STAT1: The remainder when x is divided by 7 is 2.
x = 7k + 2
Possible values of x are 2,9,16,...,51,...
we cannot say anything about the values of x
so, INSUFFICIENT

STAT2: The remainder when x is divided by 3 is 2.
x = 3s + 2
Possible values of x are 2,5,8,11,...,53,...
we cannot say anything about the values of x
so, INSUFFICIENT

Taking both together
now there are two approaches
1. write the values of t from stat1 and then from stat2 and then take the common values
from STAT1 x = 2,9,16,23,30,37,44,51,58,...,65,...
from STAT2 x = 2,5,8,...,23,...,44,...,59,65,...
common values are x = 2,23,44,65,...

2. equate x = 7k+2 to x=3s+2
we have 7k + 2 = 3s+2
k = 3s/7
since, k is an integer so only those values of s which are multiple of 7 will satisfy both STAT1 and STAT2
so, common values are given by x = 3s + 2 where s is multiple of 7
so x = 2,23,44,65 (for s=0,7,14,21 respectively)

Clearly there are values of x which are between 27 and 54 (i.e. 44) and those which are not (2,23,65)
So, both together also INSUFFICIENT

Looking for a Quant Tutor?

Check out my post for the same
starting-gmat-quant-classes-tutoring-bangalore-online-135537.html

Hope it helps!
Good Luck!
_________________
Intern  Joined: 19 Nov 2012
Posts: 1
Re: How To Solve: Remainder Problems  [#permalink]

### Show Tags

Good notes! I can certainly use them
Intern  Joined: 06 Sep 2014
Posts: 2
Re: How To Solve: Remainder Problems  [#permalink]

### Show Tags

Thank you so much, this was of great help Non-Human User Joined: 09 Sep 2013
Posts: 13719
Re: How To Solve: Remainder Problems  [#permalink]

### Show Tags

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.
_________________ Re: How To Solve: Remainder Problems   [#permalink] 09 Jan 2019, 13:02
Display posts from previous: Sort by

# How To Solve: Remainder Problems  