|
|
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
Track
Your Progress
Practice
Pays
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
45%
(medium)
Question Stats:
64% (01:24) correct
36%
(01:28)
wrong
based on 2850
sessions
History
Date
Time
Result
Not Attempted Yet
If n and m are positive integers, what is the remainder when \(3^{(4n+2)} + m\) is divided by 10?
(1) n = 2
(2) m = 1
5.gif [ 20.27 KiB | Viewed 22503 times ]
(1) n = 2
(2) m = 1
Attachment:
5.gif [ 20.27 KiB | Viewed 22503 times ]
If this question felt shaky,
try an adaptive mini quiz of similar problems in
GMAT Club Forum Quiz →. Free plan gives 5 questions per day.
Pure algebraic questions like this one are no longer part of the GMAT Data Sufficiency syllabus.
For relevant Data Sufficiency questions, please check the Data Sufficiency forum:
Data Sufficiency Forum
Data Sufficiency Forum
Given that n and m are positive integers, knowing n is useless because the expression (4n+2) will always give an answer which is 4 units apart. What we need to know is what m is for it will determine the remainder of the expression when divided by 10.
Notice that for any exponent of 3, the unit digit is repeating in cycles of 4:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
____________________________
3^5 = 243
3^6 = 729
3^7 = 2187
3^8 = 6561
____________________________
Notice the cycle: 3-9-7-1
Hence, knowing that m is 1, we know that the whole expression (4n+2)+m will give an exponent of 7, 11, 15, etc.
This ensures that the remainder will ALWAYS be 7 and B is sufficient.
do not provide the answer before it is answered, nobody may attempt it. Give it a maximum of 2 days and provide the answer thereafter.
Notice that for any exponent of 3, the unit digit is repeating in cycles of 4:
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
____________________________
3^5 = 243
3^6 = 729
3^7 = 2187
3^8 = 6561
____________________________
Notice the cycle: 3-9-7-1
Hence, knowing that m is 1, we know that the whole expression (4n+2)+m will give an exponent of 7, 11, 15, etc.
This ensures that the remainder will ALWAYS be 7 and B is sufficient.
do not provide the answer before it is answered, nobody may attempt it. Give it a maximum of 2 days and provide the answer thereafter. 
Posts: 85
mads
1: not enough as we do not know what is m hence impossible to answer remainder.
2: 3 has a cylicity of 4 i.e
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
hence cyclicity of 4
for any +ve value of n (1,2,3,.....) the unit digit of 3^(4n+2) will be 9 hence adding m=1 into it will give unit digit as 0 which is divisible by 10 hence sufficient to answer.
Its B
Posts: 99
mads
st 1) n=2, dont know about m. Not sufficient
st 2) m=1
3^(4n+2) + m = 3^[2 * (2n+1)] + m = 9 ^(2n+1) + m
2n+1 is always odd . so the units digit of 9^(2n+1) is 9 and m is 1, so the remainder will be 0
B
Geronimo
When considering the remainder when a number is divided by 10, focus on the last digit of the number. That will be the remainder. e.g. 85/10, remainder 5. 39 divided by 10, remainder 9 etc...
The last digit of powers of 3 have a cyclicity of 4. Look at the example below:
3^1 = 3
3^2 = 9
3^ 3 = 27
3^4 = 81
3^5 = 243
3^6 = 729
and so on.. notice the last digits 3, 9, 7, 1, 3, 9,....
thats the pattern they follow.
So 3^(4n + 2) will end with 9. (If you are not comfortable with cyclicity, check out its theory. You can also check out this post where I have discussed cyclicity of some other numbers in detail: https://gmatclub.com/forum/cyclicity-of-103262.html#p803511)
Now, we just need to know what m is. If m = 1, 3^(4n + 2) + m will end in 0. So remainder will be 0.
Answer (B).
_________________
-
Start FREE with selected GMAT/EA Content upon registration at anaprep.com
YouTube Channel: karishma.anaprep
Start FREE with selected GMAT/EA Content upon registration at anaprep.com
YouTube Channel: karishma.anaprep
VeritasPrepKarishma
if m=1 wouldn't it end in 7? I'm not sure how you got 0.
3^(4n + 2) + 1 = 3^(4n + 3) and based on the cyclicity pattern from above, it would be the next one over which will end in 7?
maybe i'm not understanding this correctly, please correct me. thanks!
esonrev
Joined: 15 May 2015
Last visit: 02 Mar 2022
Posts: 17
Location: United States (NY)
Schools: CBS '22 (M)
GMAT 1: 750 Q48 V45
GPA: 3.66
VeritasKarishma
I have a tough time with this not being C. Hear me out:
We will not know what the exponent will be unless we know the value of N. Exponents change the value of the integer (obviously), and thus the final digit...
3
9
27
81
243
We will not know the remainder if we don't know what N is...
what am I missing
esonrev
The actual value of n is irrelevant.
Say n = 1
3^(4n + 2) = 3^6 - The units digit here will be 9
Say n = 2
3^(4n + 2) = 3^10 - The units digit here will be 9
Say n = 3
3^(4n + 2) = 3^14 - The units digit here will be 9
and so on...
The exponent of 3 will always be of the form 4n + 2. So it will start a new cycle and end at the second term. So the units digit will always be 9. Even without the actual value of n, you know what you need to know - the units digit of 3^(4n + 2) is 9.
_________________
-
Start FREE with selected GMAT/EA Content upon registration at anaprep.com
YouTube Channel: karishma.anaprep
Start FREE with selected GMAT/EA Content upon registration at anaprep.com
YouTube Channel: karishma.anaprep
QuantMadeEasy
Joined: 27 Feb 2014
Last visit: 01 Mar 2026
Posts: 502
Location: India
Concentration: General Management, International Business
GPA: 3.97
WE:Engineering (Education)
Posts: 502
wshaffer
To find remainder when the expression is divided by 10, calculate the unit digit of the given expression
3^(4n+2) + m = 3^4n * 3^2 + m
Unit digit of 3^4n is 1
unit digit of 3^2 is 9
Require value of m to find unit digit of the expression
(1) Insufficient
(2) Sufficient
B is correct.
VeritasKarishma
Quote:
Absolutely! The "math" is to show you that cyclicality exists. Once you know it, you don't need to do it again.
3^(4n + 2) ends with a 9 for all non-negative integer values of n.
3^(4n + 1) ends with a 3 for all non-negative integer values of n.
etc
_________________
-
Start FREE with selected GMAT/EA Content upon registration at anaprep.com
YouTube Channel: karishma.anaprep
Start FREE with selected GMAT/EA Content upon registration at anaprep.com
YouTube Channel: karishma.anaprep
Kinshook
Major Poster
Joined: 03 Jun 2019
Last visit: 14 Sep 2026
Posts: 6,129
Location: India
Schools: HBS '28 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Products:
wshaffer
Asked: If n and m are positive integers, what is the remainder when \(3^{(4n+2)} + m\) is divided by 10?
(1) n = 2
\(3^{(4n+2)} + m = 3^{10} + m \)
The remainder when \(3^{(4n+2)} + m\) is divided by 10 = Remainder when m is divided by 10
Since m is unknown
NOT SUFFICIENT
(2) m = 1
\(3^{(4n+2)} + m = 3^{(4n+2)} + 1 \)
The remainder when \(3^{(4n+2)} + m\) is divided by 10
Since \(3^4n = 81^n\), the remainder when \(3^{4n}\) when divided by 10 = 1
\(3^2 = 9\)
The remainder when \(3^{(4n+2)} \)when divided by 10 = 1 * 9 = 9
The remainder when \(3^{(4n+2)} + 1\) when divided by 10 = 0
SUFFICIENT
IMO B
dylanl1218
Current Student
Joined: 20 Oct 2018
Last visit: 07 Jun 2022
Posts: 39
Location: United States
Schools: Johnson '24 (A)
GPA: 3.5
Schools: Johnson '24 (A)
Posts: 39
wshaffer
Stem Analysis:
3^4n+2 = (3^4n)(3^2)
So now we can break this down even further so that 9(3^4n)/10 + M/10. Now remember that bases to a power divided by the same denominator creates patterns. In this case any 3^4th power divided by 10 will leave some integer PLUS a remainder of 1. So 9(Q+1/10), in which Q is some integer. So now we have that 9Q + 9/10 + M/10. But we still don't know the value of M. If we have the value of M we can solve the question.
Statement 1
As analyzed in our stem the value of N doesn't help us, we need M. Not sufficient.
Statement 2
We have M, and so that means that the remainder must be zero because 9Q + 9/10 + 1/10 = 9Q + 1. Sufficient.
VeritasKarishma
3^4n+2 ,
I looked at it this way: 4n will always be even, +2 will still keep it even
so thus 4n+2 is always even. the cyclicity of 3 is 3,9,7,1. So, I inferred that last digit would be 9 or 1
So, stat 2 wasnt sufficient
Now that Im not under time pressure, I tested numbers for n and saw that it always ends up being 9. But why is that? Why does it never end in 1?
Bunuel VeritasKarishma ScottTargetTestPrep @
Chitra657
Chitra657
4n we know is an even number but we know something else about it too. That it is a multiple of 4.
Cyclicity of 3 is 4. So whenever the power is a multiple of 4, it will end in 1.
But here the power is 2 more than a multiple of 4 (it is 4n + 2) so the units digit of 3^(4n+2) will always be 9.
_________________
-
Start FREE with selected GMAT/EA Content upon registration at anaprep.com
YouTube Channel: karishma.anaprep
Start FREE with selected GMAT/EA Content upon registration at anaprep.com
YouTube Channel: karishma.anaprep
Automated notice from GMAT Club BumpBot:
A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.
This post was generated automatically.
_________________
A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.
This post was generated automatically.
_________________
Solving this question helps.
Taking a timed set of similar questions in
GMAT Club Forum Quiz →
is even better.
Moderator:
Announcements
Monday, Sep 14, 2026
11:30am NY / 3:30pm London / 9pm Mumbai
Subscribe to GMAT Club for New Podcast Episodes, Debriefs, & Insightful Sessions on GMAT Preparation
Tuck at Dartmouth
AI at Dartmouth Tuck: How the School Is Preparing Leaders for the Age of Artificial Intelligence
Inside the courses, research, and hands-on learning defining wise, decisive leadership in an AI-driven world.
Read More
Inside the courses, research, and hands-on learning defining wise, decisive leadership in an AI-driven world.
Read More
Top 5 GMAT Debrief Videos
Success stories and strategies from high-scoring candidates.