Last visit was: 14 Jul 2025, 14:50 It is currently 14 Jul 2025, 14:50
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
555-605 Level|   Remainders|      
User avatar
Bunuel
User avatar
Math Expert
Joined: 02 Sep 2009
Last visit: 14 Jul 2025
Posts: 102,570
Own Kudos:
741,404
 [35]
Given Kudos: 98,182
Products:
GMAT Club Tests Forum Quiz
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 102,570
Kudos: 741,404
 [35]
When p is divided by 7, the remainder is 4. When p is divided by 4, th 05 Nov 2017, 03:06
3
Kudos
Add Kudos
32
Bookmarks
Bookmark this Post
00:00
Start Timer
Pause Timer
Resume Timer
Show Answer
Hide
Show
History
B
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!

Difficulty:

35% (medium)

Question Stats:

75% (02:29) correct 25% (02:28) wrong based on 421 sessions

History

Date
Time
Result
Not Attempted Yet
When p is divided by 7, the remainder is 4. When p is divided by 4, the remainder is 1. How many different values of p are less than 120?

(A) 2

(B) 4

(C) 5

(D) 11

(E) 18
ShowHide Answer
Official Answer
B
Most Helpful Reply
User avatar
JeffTargetTestPrep
User avatar
Target Test Prep Representative
Joined: 04 Mar 2011
Last visit: 05 Jan 2024
Posts: 2,996
Own Kudos:
7,930
 [8]
Given Kudos: 1,646
Status:Head GMAT Instructor
Affiliations: Target Test Prep
Expert
Expert reply
Posts: 2,996
Kudos: 7,930
 [8]
When p is divided by 7, the remainder is 4. When p is divided by 4, th 08 Nov 2017, 17:32
1
Kudos
Add Kudos
7
Bookmarks
Bookmark this Post
Bunuel
When p is divided by 7, the remainder is 4. When p is divided by 4, the remainder is 1. How many different values of p are less than 120?

(A) 2

(B) 4

(C) 5

(D) 11

(E) 18

We have the following two equations:

p = 7Q + 4 where Q is a whole number.

So, p can be values such as 4, 11, 18, 25, …, 7(16) + 4 = 116.

and

p = 4S + 1 where S is a whole number.

So, p can be values such as 1, 5, 13, 14, 21, 25, …, 4(29) + 1 = 117.

We see that 25 is the first matching value of p. Since the LCM of 7 and 4 is 28, the next will be 25 + 28 = 53, and the next will be 53 + 28 = 81, and the final matching value will be 81 + 28 = 109, so there are a total of 4 values of p less than 120 that satisfy the two criteria.

Alternate Solution:

We have

p = 7Q + 4

for some Q and

p = 4S + 1

for some S.

Then, p + 3 = 7Q + 7 = 4S + 4 is divisible by both 7 and 4. Let’s list the numbers less than 120 that are divisible by 7 and 4, i.e., by 28: 0, 28, 56, 84, 112. However, these are values for p + 3; thus, the corresponding values for p are -3, 25, 53, 81, and 109. p can take any value in this list except for -3; therefore, there are four values in total.

Answer: B
General Discussion
avatar
puneetj88
avatar
Joined: 07 Jul 2015
Last visit: 19 Apr 2019
Posts: 32
Own Kudos:
16
Given Kudos: 20
Location: India
Concentration: Operations, General Management
Schools: Krannert '19 (D) Jones '19 (D) ISB '19 (S) Kellogg '20 Haas '20 Sloan '20 Smeal" 20 HBS '20
GMAT 1: 670 Q50 V30
GMAT 2: 660 Q47 V35
GPA: 3.7
WE:Operations (Manufacturing)
Products:
My Reviews
Schools: Krannert '19 (D) Jones '19 (D) ISB '19 (S) Kellogg '20 Haas '20 Sloan '20 Smeal" 20 HBS '20
GMAT 2: 660 Q47 V35
Posts: 32
Kudos: 16
When p is divided by 7, the remainder is 4. When p is divided by 4, th 05 Nov 2017, 03:52
Kudos
Add Kudos
Bookmarks
Bookmark this Post
ONLY AN ODD NUMBER WILL LEAVE REMAINDER 1 WHEN DIVIDED BY 4, SO WE ACTUALLY WANT ODD NUMBERS WHICH WILL LEAVE REMAINDER 4 WHEN DIVIDED BY 7 AND REMAINDER 1 WHEN DIVIDED BY 4.
25,53,81,109

ANS:B
User avatar
chetan2u
User avatar
GMAT Expert
Joined: 02 Aug 2009
Last visit: 14 Jul 2025
Posts: 11,294
Own Kudos:
41,743
 [4]
Given Kudos: 333
Status:Math and DI Expert
Products:
My Reviews
Expert
Expert reply
Posts: 11,294
Kudos: 41,743
 [4]
When p is divided by 7, the remainder is 4. When p is divided by 4, th 05 Nov 2017, 04:22
2
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
Bunuel
When p is divided by 7, the remainder is 4. When p is divided by 4, the remainder is 1. How many different values of p are less than 120?

(A) 2

(B) 4

(C) 5

(D) 11

(E) 18


Hi..

an algebraic method..

p is divided by 7, the remainder is 4 MEANS \(p = 7x+4\)
p is divided by 4, the remainder is 1 MEANS \(p = 4y+1\)

so \(7x+4=4y+1.......7x+4-1=4y...........7x+3=4y\)
so 7x+3 should be a MULTIPLE of 4 as RHS is 4y..
first value of x as 3 fits in 7x+3=7*3+3=24..
next value of x will be 3+4 and then 3+2*4 and so on.. formula = 3+4n where n is 0,1,2,3.....

the equation as per restriction of <120
\(7x+4<120.......7(3+4n)<120.......21+28n<120.........28n<99..........n<\frac{99}{28}\).
\(n<3.abc\)
so n can take values from 0 to 3 : 0,1,2,3...... 4 values

or else, after finding first value as 25 keep adding 28 to it till you get >120
B
User avatar
frndrobinster
User avatar
Joined: 02 Sep 2015
Last visit: 11 Nov 2017
Posts: 31
Own Kudos:
68
 [4]
Given Kudos: 21
Location: India
Concentration: General Management, Entrepreneurship
Schools: Sauder 'Dec 18
GPA: 3.79
WE:General Management (Energy)
Schools: Sauder 'Dec 18
Posts: 31
Kudos: 68
 [4]
When p is divided by 7, the remainder is 4. When p is divided by 4, th 05 Nov 2017, 04:29
3
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Let the number be 7a+4 = 4b+1 ( When p is divided by 7, the remainder is 4. When p is divided by 4, the remainder is 1)
Equating we get, 4b-7a = 3
Solving for integral value of a, b ; we get min value of (a, b) = (3, 6)
hence number is 7*3+4 = 25
the smallest number is 25
for other numbers add (LCM of 7 & 4)28 to 25 to get other numbers. hence others nos are 25, 53,81,109,137 and so on...
Numbers smaller than 120 will be 25, 53,81,109.
hence there will be 4 such numbers.
User avatar
gracie
User avatar
Joined: 07 Dec 2014
Last visit: 11 Oct 2020
Posts: 1,035
Own Kudos:
1,860
 [4]
Given Kudos: 27
Posts: 1,035
Kudos: 1,860
 [4]
When p is divided by 7, the remainder is 4. When p is divided by 4, th 05 Nov 2017, 14:28
1
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
Bunuel
When p is divided by 7, the remainder is 4. When p is divided by 4, the remainder is 1. How many different values of p are less than 120?

(A) 2

(B) 4

(C) 5

(D) 11

(E) 18

let quotients=x and y
p-4=7x
p-1=4y
7x=4y-3
after testing 0,1, and 2 for x,
least possible values for x and y are 3 and 6 respectively
least value of p=25
25+(n-1)(7*4)<120
28n<123
n<4.4
n=4
B
avatar
Psk13
avatar
Joined: 23 Jul 2016
Last visit: 15 Nov 2021
Posts: 21
Own Kudos:
7
Given Kudos: 176
Posts: 21
Kudos: 7
When p is divided by 7, the remainder is 4. When p is divided by 4, th 18 Nov 2017, 05:53
Kudos
Add Kudos
Bookmarks
Bookmark this Post
JeffTargetTestPrep
Bunuel
When p is divided by 7, the remainder is 4. When p is divided by 4, the remainder is 1. How many different values of p are less than 120?

(A) 2

(B) 4

(C) 5

(D) 11

(E) 18

We have the following two equations:

p = 7Q + 4 where Q is a whole number.

So, p can be values such as 4, 11, 18, 25, …, 7(16) + 4 = 116.

and

p = 4S + 1 where S is a whole number.

So, p can be values such as 1, 5, 13, 14, 21, 25, …, 4(29) + 1 = 117.

We see that 25 is the first matching value of p. Since the LCM of 7 and 4 is 28, the next will be 25 + 28 = 53, and the next will be 53 + 28 = 81, and the final matching value will be 81 + 28 = 109, so there are a total of 4 values of p less than 120 that satisfy the two criteria.

Alternate Solution:

We have

p = 7Q + 4

for some Q and

p = 4S + 1

for some S.

Then, p + 3 = 7Q + 7 = 4S + 4 is divisible by both 7 and 4. Let’s list the numbers less than 120 that are divisible by 7 and 4, i.e., by 28: 0, 28, 56, 84, 112. However, these are values for p + 3; thus, the corresponding values for p are -3, 25, 53, 81, and 109. p can take any value in this list except for -3; therefore, there are four values in total.

Answer: B


hi
your first method is quite interesting
i want to ask is this some kind of rule/ some basic knowledge (adding the lcm to the first common number to get other possibilities)
it can be applicable to every similar problem
and
i didnt get the hang of your second method
can you please elaborate it a bit
thanku
your
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 14 Jul 2025
Posts: 16,106
Own Kudos:
74,320
 [3]
Given Kudos: 475
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,106
Kudos: 74,320
 [3]
When p is divided by 7, the remainder is 4. When p is divided by 4, th 11 Feb 2019, 23:07
2
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
Bunuel
When p is divided by 7, the remainder is 4. When p is divided by 4, the remainder is 1. How many different values of p are less than 120?

(A) 2

(B) 4

(C) 5

(D) 11

(E) 18

When divided by 7, the remainder is 4 (Same as -3)
When divided by 4, the remainder is 1 (Same as -3)

Common remainder = -3

p = 28m - 3
All numbers of this format will satisfy these conditions.
m can take 4 values: 1, 2, 3, 4

Answer (B)

Here is a bit about negative remainders and their application: https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2014/0 ... -the-gmat/
User avatar
BrushMyQuant
User avatar
Joined: 05 Apr 2011
Last visit: 13 Jul 2025
Posts: 2,247
Own Kudos:
2,462
Given Kudos: 100
Status:Tutor - BrushMyQuant
Location: India
Concentration: Finance, Marketing
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
GPA: 3
WE:Information Technology (Computer Software)
Expert
Expert reply
Schools: XLRI (A)
GMAT 1: 700 Q51 V31
Posts: 2,247
Kudos: 2,462
When p is divided by 7, the remainder is 4. When p is divided by 4, th 20 Oct 2022, 09:59
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Given that When p is divided by 7, the remainder is 4. When p is divided by 4, the remainder is 1. And we need to find How many different values of p are less than 120

Theory: Dividend = Divisor*Quotient + Remainder

When p is divided by 7, the remainder is 4

p -> Dividend
7 -> Divisor
a -> Quotient (Assume)
4 -> Remainders
=> p = 7*a + 4 = 7a + 4

When p is divided by 4, the remainder is 1

p -> Dividend
4 -> Divisor
b -> Quotient (Assume)
1 -> Remainders
=> p = 4*b + 1 = 4b + 1

p = 7a + 4 = 4b + 1
=> b = \(\frac{7a + 3}{4}\)

Only those values of "a" which will also give "b" as integer will give us the common values of p
a = 3, 7, 11, 15, 19,...
But a = 19 will give us a value of > 120 for p

=> 4 value of p which are less than 120 are possible

So, Answer will be B
Hope it helps!

Watch the following video to learn the Basics of Remainders

User avatar
martinfzb
User avatar
Joined: 05 Sep 2022
Last visit: 19 Dec 2022
Posts: 24
Own Kudos:
16
Given Kudos: 1
Posts: 24
Kudos: 16
When p is divided by 7, the remainder is 4. When p is divided by 4, th 24 Oct 2022, 08:28
Kudos
Add Kudos
Bookmarks
Bookmark this Post
When time is short, we can list out the value to find some logic.

N 7*n+4 4*m+1
0 4 1
1 11 5
2 18 9
3 25 13
4 32 17
5 39 21
6 46 25
7 53 29
8 61 33
9 60 37
10 74 41
11 81 45
12 88 49
13 95 53
while n =3 and m=6, 7*3+4=25=4*6+1
and while n=7 and m=13, 7*7+4=53=4*13+1
the logix is that
n increase at a step of 7-3=4,
m increase at a step of 13-6=7
120 /7 = 17 and 17/4= 4 step at most
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 37,398
Own Kudos:
1,013
Posts: 37,398
Kudos: 1,013
When p is divided by 7, the remainder is 4. When p is divided by 4, th 13 Jul 2025, 09:50
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.
Moderators:
Bunuel
Math Expert
102570 posts
Krunaal
PS Forum Moderator
691 posts