Last visit was: 19 Nov 2025, 06:04 It is currently 19 Nov 2025, 06:04
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
605-655 Level|   Exponents|   Remainders|      
User avatar
wshaffer
User avatar
Joined: 12 Jul 2006
Last visit: 24 Oct 2008
Posts: 84
Own Kudos:
266
 [167]
Posts: 84
Kudos: 266
 [167]
If n and m are positive integers, what is the remainder when 3^(4n+2) Updated on: 12 Aug 2022, 06:27
Originally posted by wshaffer on 12 Nov 2006, 14:38.
Last edited by Bunuel on 12 Aug 2022, 06:27, edited 2 times in total.
Renamed the topic and edited the question.
9
Kudos
Add Kudos
156
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:

45% (medium)

Question Stats:

64% (01:23) correct 36% (01:28) wrong based on 2563 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


Show Spoiler
Attachment:
5.gif
5.gif [ 20.27 KiB | Viewed 20064 times ]
ShowHide Answer
Official Answer
B
Most Helpful Reply
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 18 Nov 2025
Posts: 16,267
Own Kudos:
76,993
 [62]
Given Kudos: 482
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,267
Kudos: 76,993
 [62]
If n and m are positive integers, what is the remainder when 3^(4n+2) 18 Nov 2010, 07:58
26
Kudos
Add Kudos
34
Bookmarks
Bookmark this Post
Geronimo
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
Show more

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).
User avatar
bangalorian2000
User avatar
Joined: 01 Feb 2010
Last visit: 05 Jul 2011
Posts: 85
Own Kudos:
152
 [7]
Given Kudos: 2
Posts: 85
Kudos: 152
 [7]
If n and m are positive integers, what is the remainder when 3^(4n+2) 06 Apr 2010, 21:31
5
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
mads
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
Show more

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
General Discussion
User avatar
Paul
User avatar
Joined: 15 Dec 2003
Last visit: 10 Nov 2012
Posts: 2,708
Own Kudos:
1,630
 [4]
Posts: 2,708
Kudos: 1,630
 [4]
If n and m are positive integers, what is the remainder when 3^(4n+2) 12 Nov 2006, 15:17
1
Kudos
Add Kudos
3
Bookmarks
Bookmark this Post
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.

:btw 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.
User avatar
chix475ntu
User avatar
Joined: 26 May 2005
Last visit: 03 Aug 2011
Posts: 99
Own Kudos:
282
 [4]
Given Kudos: 1
Posts: 99
Kudos: 282
 [4]
If n and m are positive integers, what is the remainder when 3^(4n+2) 06 Apr 2010, 22:48
3
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
mads
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
Show more
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
User avatar
gtr022001
User avatar
Joined: 07 Jan 2010
Last visit: 20 Oct 2013
Posts: 73
Own Kudos:
361
Given Kudos: 57
Location: So. CA
Concentration: General
WE 1: 2 IT
WE 2: 4 Software Analyst
Posts: 73
Kudos: 361
If n and m are positive integers, what is the remainder when 3^(4n+2) 20 Nov 2010, 19:38
Kudos
Add Kudos
Bookmarks
Bookmark this Post
VeritasPrepKarishma
Geronimo
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

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).
Show more

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!
User avatar
esonrev
User avatar
Joined: 15 May 2015
Last visit: 02 Mar 2022
Posts: 17
Own Kudos:
11
Given Kudos: 269
Location: United States (NY)
Schools: CBS '22 (M)
GMAT 1: 750 Q48 V45
GPA: 3.66
Schools: CBS '22 (M)
GMAT 1: 750 Q48 V45
Posts: 17
Kudos: 11
If n and m are positive integers, what is the remainder when 3^(4n+2) 28 Nov 2019, 11:11
Kudos
Add Kudos
Bookmarks
Bookmark this Post
VeritasKarishma
Geronimo
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

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).
Show more

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
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 18 Nov 2025
Posts: 16,267
Own Kudos:
76,993
 [6]
Given Kudos: 482
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,267
Kudos: 76,993
 [6]
If n and m are positive integers, what is the remainder when 3^(4n+2) 01 Dec 2019, 21:31
4
Kudos
Add Kudos
2
Bookmarks
Bookmark this Post
esonrev
VeritasKarishma
Geronimo
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

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).

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
Show more

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.
User avatar
QuantMadeEasy
User avatar
Joined: 28 Feb 2014
Last visit: 15 Nov 2025
Posts: 502
Own Kudos:
785
 [1]
Given Kudos: 78
Location: India
Concentration: General Management, International Business
GPA: 3.97
WE:Engineering (Education)
Posts: 502
Kudos: 785
 [1]
If n and m are positive integers, what is the remainder when 3^(4n+2) 01 Dec 2019, 22:20
Kudos
Add Kudos
1
Bookmarks
Bookmark this Post
wshaffer
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
Show more
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.
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 18 Nov 2025
Posts: 16,267
Own Kudos:
76,993
Given Kudos: 482
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,267
Kudos: 76,993
If n and m are positive integers, what is the remainder when 3^(4n+2) 18 Mar 2020, 21:34
Kudos
Add Kudos
Bookmarks
Bookmark this Post
VeritasKarishma
Geronimo
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

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).
Show more

Quote:

Without doing all the math by trying different n values. Could you assume the exponent will always be the second one in the cyclicity because of the +2 in the exponent?
To better explain, if the exponent was (4n+1), could we assume that the units digit would be 3?
Show more

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
User avatar
Kinshook
User avatar
Major Poster
Joined: 03 Jun 2019
Last visit: 18 Nov 2025
Posts: 5,794
Own Kudos:
5,509
Given Kudos: 161
Location: India
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
WE:Engineering (Transportation)
Products:
My Reviews
Schools: HBS '26 CBS '26 Wharton '26
GMAT 1: 690 Q50 V34
Posts: 5,794
Kudos: 5,509
If n and m are positive integers, what is the remainder when 3^(4n+2) 02 Aug 2020, 00:00
Kudos
Add Kudos
Bookmarks
Bookmark this Post
wshaffer
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
Show more

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
User avatar
dylanl1218
User avatar
Current Student
Joined: 20 Oct 2018
Last visit: 07 Jun 2022
Posts: 40
Own Kudos:
10
Given Kudos: 246
Location: United States
Schools: Johnson '24 (A)
GPA: 3.5
Schools: Johnson '24 (A)
Posts: 40
Kudos: 10
If n and m are positive integers, what is the remainder when 3^(4n+2) 21 Jul 2021, 12:55
Kudos
Add Kudos
Bookmarks
Bookmark this Post
wshaffer
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
Show more

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.
avatar
Chitra657
avatar
Joined: 18 May 2021
Last visit: 15 Mar 2022
Posts: 41
Own Kudos:
7
Given Kudos: 51
Posts: 41
Kudos: 7
If n and m are positive integers, what is the remainder when 3^(4n+2) 28 Oct 2021, 07:05
Kudos
Add Kudos
Bookmarks
Bookmark this Post
VeritasKarishma
Geronimo
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

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).
Show more

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 @
User avatar
KarishmaB
User avatar
Joined: 16 Oct 2010
Last visit: 18 Nov 2025
Posts: 16,267
Own Kudos:
76,993
 [1]
Given Kudos: 482
Location: Pune, India
Expert
Expert reply
Active GMAT Club Expert! Tag them with @ followed by their username for a faster response.
Posts: 16,267
Kudos: 76,993
 [1]
If n and m are positive integers, what is the remainder when 3^(4n+2) 28 Oct 2021, 20:53
1
Kudos
Add Kudos
Bookmarks
Bookmark this Post
Chitra657
VeritasKarishma
Geronimo
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

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).

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 @
Show more

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.
User avatar
bumpbot
User avatar
Non-Human User
Joined: 09 Sep 2013
Last visit: 04 Jan 2021
Posts: 38,586
Own Kudos:
1,079
Posts: 38,586
Kudos: 1,079
If n and m are positive integers, what is the remainder when 3^(4n+2) 19 Oct 2024, 23:55
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
105388 posts
kingbucky
496 posts