What is the remainder of (3^7^11)/5 : GMAT Problem Solving (PS)
Check GMAT Club Decision Tracker for the Latest School Decision Releases http://gmatclub.com/AppTrack

 It is currently 23 Jan 2017, 05:46

### 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

# Events & Promotions

###### Events & Promotions in June
Open Detailed Calendar

# What is the remainder of (3^7^11)/5

Author Message
TAGS:

### Hide Tags

Director
Joined: 03 Sep 2006
Posts: 879
Followers: 6

Kudos [?]: 772 [0], given: 33

What is the remainder of (3^7^11)/5 [#permalink]

### Show Tags

24 Jan 2012, 07:22
15
This post was
BOOKMARKED
00:00

Difficulty:

75% (hard)

Question Stats:

46% (02:09) correct 54% (00:56) wrong based on 726 sessions

### HideShow timer Statistics

What is the remainder of (3^7^11)/5

A. 0
B. 1
C. 2
D. 3
E. 4

Please explain what should be the approach in such kind of questions?

Also, someone please tell how I could use the formatting to display 3 raised to power 7 raised to power 11, divided by 5?
[Reveal] Spoiler: OA
Manager
Status: Employed
Joined: 17 Nov 2011
Posts: 100
Location: Pakistan
GMAT 1: 720 Q49 V40
GPA: 3.2
WE: Business Development (Internet and New Media)
Followers: 7

Kudos [?]: 135 [8] , given: 10

Re: PS-What is the remainder [#permalink]

### Show Tags

24 Jan 2012, 07:43
8
KUDOS
1
This post was
BOOKMARKED
The answer should be C i.e. 2.

Here is how.

The expression is : $$3^{7^{11}}$$

Lets first resolve $$7^{11}$$

So $$7^2=49$$

So $$7^4=.....1$$ where 1 is the last digit of the number

So $$7^8=.....1$$ where 1 is the last digit of the number

So $$7^3=.....3$$ where 3 is the last digit of the number

So $$7^{11}=7^3*7^8=........1*3=.............3$$ where 3 is the last digit of the number

Now on to $$3^{7^{11}}=3^{......3}$$ where 3 is the last digit of the exponent

We know the power is odd and its 3. Lets check out the last digit of some of the odd exponents for $$3$$

So $$3^1=3$$
So $$3^3=27$$
& $$3^5=....3$$
& $$3^7=.....7$$
& $$3^{11}=.........7$$

Recognize the patterns. It could be 3 or 7. Also, it is is 3 only in the case when the exponent is $$1$$ or a multiple of $$5$$

We know the power is greater than $$1$$ and not a multiple of $$5$$ so the only possibility for the last digit is $$7$$

Now we know that $$\frac{7}{5}$$ remainder is $$2$$ Hence the answer must be C.
_________________

"Nowadays, people know the price of everything, and the value of nothing." Oscar Wilde

Intern
Joined: 23 Jan 2012
Posts: 7
Location: India
Concentration: Strategy, Finance
GMAT 1: 620 Q51 V23
GPA: 3.29
WE: Engineering (Other)
Followers: 0

Kudos [?]: 16 [2] , given: 1

Re: PS-What is the remainder [#permalink]

### Show Tags

24 Jan 2012, 11:31
2
KUDOS
omerrauf wrote:
The answer should be C i.e. 2.

Here is how.

The expression is : $$3^{7^{11}}$$

Lets first resolve $$7^{11}$$

So $$7^2=49$$

So $$7^4=.....1$$ where 1 is the last digit of the number

So $$7^8=.....1$$ where 1 is the last digit of the number

So $$7^3=.....3$$ where 3 is the last digit of the number

So $$7^{11}=7^3*7^8=........1*3=.............3$$ where 3 is the last digit of the number

Now on to $$3^{7^{11}}=3^{......3}$$ where 3 is the last digit of the exponent

We know the power is odd and its 3. Lets check out the last digit of some of the odd exponents for $$3$$

So $$3^1=3$$
So $$3^3=27$$
& $$3^5=....3$$
& $$3^7=.....7$$
& $$3^{11}=.........7$$

Recognize the patterns. It could be 3 or 7. Also, it is is 3 only in the case when the exponent is $$1$$ or a multiple of $$5$$

We know the power is greater than $$1$$ and not a multiple of $$5$$ so the only possibility for the last digit is $$7$$

Now we know that $$\frac{7}{5}$$ remainder is $$2$$ Hence the answer must be C.

Well the answer posted above is correct, but the reasoning provided above is a little bit flawed.
If you look at $$3^{13}$$ it has '3' as the units digit in exponent and if you try to solve it the answer will have 3 as the units digit, which when divided by 5 will give 3 as the remainder.

Here's another approach which I believe you can use
Understand that in this question all you need to find out is the units digit of the expression $$3^{7^{11}}$$. In order to do so, you must reduce this term in the form of $$3^x$$.
$$3^x$$ has a cyclicity of 4, i.e. the units digit of $$3^x$$ repeats itself after four terms
$$3^1=3$$ --- so --- $$3^{4n+1}=...3$$------------------equation (1)
$$3^2=9$$ --- so --- $$3^{4n+2}=...9$$------------------equation (2)
$$3^3=27$$ --- so --- $$3^{4n+3}=...7$$------------------equation (3)
$$3^4=81$$ --- so --- $$3^{4n+4}=...1$$------------------equation (4)
$$3^5=243$$ --- so --- $$3^5=3^{4+1}=3^{4n+1}$$
So we can write any power of 3 in the form of $$3^{4n+k}$$. This way calculating the value of 'k', we can easily find the units digit.
Here also we just need to write the power of 3 in 4n+k form.
Lets concentrate on $$7^{11}$$
If we divide this by 4, whatever we get will be the value of 'k' and our problem would be solved. Rewriting it as $$(8-1)^{11}$$
Divide $$(8-1)^{11}$$ by 4 to get the value of k.
Here 8 will give the remainder 0 when divided by 4 and the only remainder we will get is from -1. Using the concept of negative remainders(which I'm assuming you know, incase you don't, feel free to ping me and I'll be happy to tell you) we'll get 3 as our final remainder
Hence $$7^{11}$$ can be written as 4n+3.
So we can write our given expression $$3^{7^{11}}$$ as $$3^{4n+3}$$.
Using equation (3) above, we can easily make out that our units digit will be 7. Dividing this by 5 will give 2 as the remainder.

PS: Forgive me for my poor formatting. I'm still learning

Hope this helps.
Math Expert
Joined: 02 Sep 2009
Posts: 36609
Followers: 7099

Kudos [?]: 93511 [2] , given: 10568

Re: PS-What is the remainder [#permalink]

### Show Tags

24 Jan 2012, 12:15
2
KUDOS
Expert's post
4
This post was
BOOKMARKED
LM wrote:
What is the remainder of (3^7^11)/5

A. 0
B. 1
C. 2
D. 3
E. 4

Please explain what should be the approach in such kind of questions?

Also, someone please tell how I could use the formatting to display 3 raised to power 7 raised to power 11, divided by 5?

First of all I think that this question is a little bit out of the scope of the GMAT. But anyway:

The last digit of 3 in positive integer power repeats in pattern of 4: {3, 9, 7, 1}. So, basically we should find the remainder upon division 7^(11) by cyclicity of 4 (to see on which number in this pattern $$7^{11}$$ falls on). $$7^{11}=(4+3)^{11}$$, now if we expand this expression all terms but the last one will have 4 in them, thus will leave no remainder upon division by 4, the last term will be $$3^{11}$$. Thus the question becomes: what is the remainder upon division $$3^{11}$$ by 4:
3 divided by 4 yields remainder of 3;
3^2=9 divided by 4 yields remainder of 1;
3^3=27 divided by 4 yields remainder of 3;
3^4=81 divided by 4 yields remainder of 1.

So, 3 in odd power yields remainder of 1 upon division by 4 --> $$3^{11}$$ yields remainder of 3 --> finally, we have that $$3^{7^{11}}$$ will have the same last digit as $$3^3$$, which is 7. Thus as $$3^{7^{11}}$$ has the last digit of 7 then divided by 5 it will yield remainder of 2.

LM wrote:
Also, someone please tell how I could use the formatting to display 3 raised to power 7 raised to power 11, divided by 5?

Mark \frac{3^{7^{11}}}{5} and press (m) button: $$\frac{3^{7^{11}}}{5}$$
_________________
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7130
Location: Pune, India
Followers: 2140

Kudos [?]: 13702 [1] , given: 222

Re: PS-What is the remainder [#permalink]

### Show Tags

26 Jan 2012, 00:41
1
KUDOS
Expert's post
2
This post was
BOOKMARKED
LM wrote:

Helps but can you give some link or details about concept of negative remainders.

Here are some links that discuss divisibility and remainders. The third link discusses negative remainders but I think it would make more sense if you first go through the first two links.

http://www.veritasprep.com/blog/2011/04 ... unraveled/
http://www.veritasprep.com/blog/2011/04 ... y-applied/
http://www.veritasprep.com/blog/2011/05 ... emainders/
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Math Expert Joined: 02 Sep 2009 Posts: 36609 Followers: 7099 Kudos [?]: 93511 [0], given: 10568 Re: PS-What is the remainder [#permalink] ### Show Tags 26 Jan 2012, 03:52 LM wrote: Helps but can you give some link or details about concept of negative remainders. If you are preparing for the GMAT you probably shouldn't waste you valuable time on the out of the scope questions like this or on the concepts that aren't tested. _________________ Intern Joined: 23 Jan 2012 Posts: 7 Location: India Concentration: Strategy, Finance GMAT 1: 620 Q51 V23 GPA: 3.29 WE: Engineering (Other) Followers: 0 Kudos [?]: 16 [0], given: 1 Re: What is the remainder of (3^7^11)/5 [#permalink] ### Show Tags 26 Jan 2012, 08:05 Links given by Karishma should help you. Are you preparing for CAT? Because I don't think GMAT tests you on such difficult questions. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7130 Location: Pune, India Followers: 2140 Kudos [?]: 13702 [0], given: 222 Re: What is the remainder of (3^7^11)/5 [#permalink] ### Show Tags 26 Jan 2012, 21:11 Expert's post 1 This post was BOOKMARKED Such questions are definitely not GMAT's style but the concept of negative remainders is interesting and useful in certain situations. Besides its good to understand it as a part of the theory of divisibility and remainders. It is the complementary concept of positive remainders. So go ahead and check out the posts. They are all GMAT relevant. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Math Expert
Joined: 02 Sep 2009
Posts: 36609
Followers: 7099

Kudos [?]: 93511 [0], given: 10568

Re: What is the remainder of (3^7^11)/5 [#permalink]

### Show Tags

26 Jan 2012, 23:46
VeritasPrepKarishma wrote:
Such questions are definitely not GMAT's style but the concept of negative remainders is interesting and useful in certain situations. Besides its good to understand it as a part of the theory of divisibility and remainders. It is the complementary concept of positive remainders. So go ahead and check out the posts. They are all GMAT relevant.

I agree with Karishma that the concept of negative remainders is interesting and useful in certain situations. Having said that I'd like to point out two issues:

1. Every divisibility/remainder question on the GMAT can be (easily) solved without this concept;

2. General/common definition of a remainder is that it's more than or equal to zero and less than divisor. Reffer to OG12:

If x and y are positive integers, there exist unique integers q and r, called the quotient and remainder,
respectively, such that y = xq + r and 0 <= r < x.

Therefore, if time is an issue in your preparation, you should probably skip this concept (even though it's not hard at all) and concentrate more on orthodox approaches.
_________________
Intern
Joined: 12 Mar 2012
Posts: 16
Followers: 0

Kudos [?]: 7 [0], given: 19

Re: What is the remainder of (3^7^11)/5 [#permalink]

### Show Tags

25 Mar 2012, 10:09
I solved this 1 as follows:

3^7^11/5=???

7^11= 7^3 * 7^3* 7^3*7^2= 343*343*343*49

Last digit of above will be 3*3*3*9 = 3

eqn reduces to 3^3/5= 27/5=2

Guys , havse solved the questn with unit digit method.

Please correct me if i m wrong.

If i m rite, i deserve the kudo !!!
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 7130
Location: Pune, India
Followers: 2140

Kudos [?]: 13702 [0], given: 222

Re: What is the remainder of (3^7^11)/5 [#permalink]

### Show Tags

25 Mar 2012, 20:22
Expert's post
1
This post was
BOOKMARKED
Cmplkj123 wrote:
I solved this 1 as follows:

3^7^11/5=???

7^11= 7^3 * 7^3* 7^3*7^2= 343*343*343*49

Last digit of above will be 3*3*3*9 = 3

eqn reduces to 3^3/5= 27/5=2

Guys , havse solved the questn with unit digit method.

Please correct me if i m wrong.

If i m rite, i deserve the kudo !!!

Actually, the logic is not entirely correct.
$$3^{ab...3}$$ i.e. 3 to a power that ends in 3 will not necessarily give you a remainder of 2 when divided by 5.
e.g. $$3^{13}$$ when divided by 5 gives 3 as the remainder.
$$3^{33}$$ when divided by 5 gives 3 as the remainder.
etc

You need to find the unit's digit of $$3^x$$ where $$x = 7^{11}$$.
Since 3 has a cyclicity of 4, you need to figure out the remainder when x is divided by 4.
$$x = 7^{11} = (8-1)^{11}$$ so remainder will be -1 i.e. 3
(for explanation of this, check out: http://www.veritasprep.com/blog/2011/05 ... ek-in-you/

So basically we have $$3^{4m + 3}$$. Since 3 has a cyclicity of 4 {3, 9, 7, 1} , the unit's digit here will be 7.
When you divide this by 5, the remainder will be 2.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 22 Jan 2012 Posts: 22 Followers: 0 Kudos [?]: 37 [0], given: 11 Re: What is the remainder of (3^7^11)/5 [#permalink] ### Show Tags 26 Mar 2012, 01:29 phew thanks so much for reassuring that this is not gmat type material i was working up a sweat just reading the explanations! Manager Joined: 14 Nov 2008 Posts: 70 Followers: 2 Kudos [?]: 23 [0], given: 1 Re: What is the remainder of (3^7^11)/5 [#permalink] ### Show Tags 30 Aug 2013, 22:54 This is basically 2 remainder questions merged in one: Part 1:Rem(7^11)/4 = Rem(8-1)^11/4=3 Part 2:Rem((3^(7^11))/5)=Rem((3^3)81^k)/5=Rem(27/5)=2 We are considering the remainder for 4 as 3^4 = 81 which can be broken into 80+1, yielding a remainder of 1. Manager Status: Persevering Joined: 15 May 2013 Posts: 225 Location: India Concentration: Technology, Leadership GMAT Date: 08-02-2013 GPA: 3.7 WE: Consulting (Consulting) Followers: 1 Kudos [?]: 86 [0], given: 34 Re: What is the remainder of (3^7^11)/5 [#permalink] ### Show Tags 08 Sep 2013, 03:54 C; 2 is the remainder 3/5^7^11=> (3^2*3^2*3^2*3 /5)^11=>(1*1*1*3)^11 /5 => as cycle for 3 repeats after 4 times =>3^3/5=> 27/5 => 2 as remainder. _________________ --It's one thing to get defeated, but another to accept it. Manager Status: folding sleeves up Joined: 26 Apr 2013 Posts: 157 Location: India Concentration: Finance, Strategy GMAT 1: 530 Q39 V23 GMAT 2: 560 Q42 V26 GPA: 3.5 WE: Consulting (Computer Hardware) Followers: 1 Kudos [?]: 85 [0], given: 39 Re: PS-What is the remainder [#permalink] ### Show Tags 23 Sep 2013, 11:11 VeritasPrepKarishma wrote: LM wrote: Helps but can you give some link or details about concept of negative remainders. Here are some links that discuss divisibility and remainders. The third link discusses negative remainders but I think it would make more sense if you first go through the first two links. http://www.veritasprep.com/blog/2011/04 ... unraveled/ http://www.veritasprep.com/blog/2011/04 ... y-applied/ http://www.veritasprep.com/blog/2011/05 ... emainders/ Hi Karishma, Here is what I did...but couldn't conclude to the correct answer. Please check the below procedure a^b^c= a^bc so 3^7^11=>3^77 (5-2)^77/5 all I have to worry is about (-2)^77 now for every 2^4 i have remainder 1 so finally i have (-2)/5 (since 77/4 remainder is 1) ....how to solve further. Please assist. Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 7130 Location: Pune, India Followers: 2140 Kudos [?]: 13702 [0], given: 222 Re: PS-What is the remainder [#permalink] ### Show Tags 23 Sep 2013, 20:09 email2vm wrote: VeritasPrepKarishma wrote: LM wrote: Helps but can you give some link or details about concept of negative remainders. Here are some links that discuss divisibility and remainders. The third link discusses negative remainders but I think it would make more sense if you first go through the first two links. http://www.veritasprep.com/blog/2011/04 ... unraveled/ http://www.veritasprep.com/blog/2011/04 ... y-applied/ http://www.veritasprep.com/blog/2011/05 ... emainders/ Hi Karishma, Here is what I did...but couldn't conclude to the correct answer. Please check the below procedure a^b^c= a^bc so 3^7^11=>3^77 (5-2)^77/5 all I have to worry is about (-2)^77 now for every 2^4 i have remainder 1 so finally i have (-2)/5 (since 77/4 remainder is 1) ....how to solve further. Please assist. The first step is incorrect: a^b^c is not equal to a^bc $$3^{7^{11}}$$ is not $$3^{77}$$ Just like 2^4 is not 2*4, $$7^{11}$$ is not 77. $$7^{11}$$ is much much greater than 77 _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Math Expert
Joined: 02 Sep 2009
Posts: 36609
Followers: 7099

Kudos [?]: 93511 [0], given: 10568

Re: PS-What is the remainder [#permalink]

### Show Tags

24 Sep 2013, 06:30
email2vm wrote:
VeritasPrepKarishma wrote:
LM wrote:

Helps but can you give some link or details about concept of negative remainders.

Here are some links that discuss divisibility and remainders. The third link discusses negative remainders but I think it would make more sense if you first go through the first two links.

http://www.veritasprep.com/blog/2011/04 ... unraveled/
http://www.veritasprep.com/blog/2011/04 ... y-applied/
http://www.veritasprep.com/blog/2011/05 ... emainders/

Hi Karishma,

Here is what I did...but couldn't conclude to the correct answer. Please check the below procedure

a^b^c= a^bc so 3^7^11=>3^77

(5-2)^77/5

all I have to worry is about (-2)^77

now for every 2^4 i have remainder 1 so finally i have (-2)/5 (since 77/4 remainder is 1)

....how to solve further. Please assist.

$$(a^m)^n=a^{mn}$$

$$a^m^n=a^{(m^n)}$$ and not $$(a^m)^n$$ (if exponentiation is indicated by stacked symbols, the rule is to work from the top down).

Theory on Exponents: math-number-theory-88376.html

All DS Exponents questions to practice: search.php?search_id=tag&tag_id=39
All PS Exponents questions to practice: search.php?search_id=tag&tag_id=60

Tough and tricky DS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125967.html
Tough and tricky PS exponents and roots questions with detailed solutions: tough-and-tricky-exponents-and-roots-questions-125956.html

_________________
MBA Section Director
Status: On vacation...
Affiliations: GMAT Club
Joined: 21 Feb 2012
Posts: 3981
Location: India
City: Pune
GMAT 1: 680 Q49 V34
GPA: 3.4
Followers: 397

Kudos [?]: 2888 [1] , given: 2163

Re: PS-What is the remainder [#permalink]

### Show Tags

24 Sep 2013, 13:10
1
KUDOS
Expert's post
email2vm wrote:
Hi Karishma,

Here is what I did...but couldn't conclude to the correct answer. Please check the below procedure

a^b^c= a^bc so 3^7^11=>3^77

(5-2)^77/5

all I have to worry is about (-2)^77

now for every 2^4 i have remainder 1 so finally i have (-2)/5 (since 77/4 remainder is 1)

....how to solve further. Please assist.

Let me try to explain......................

What is the remainder of $$\frac{3^{7^{11}}}{5}$$

Remainder of $$\frac{3^1}{5}$$ is 3

Remainder of $$\frac{3^2}{5}$$ is 4

Remainder of $$\frac{3^3}{5}$$ is 2

Remainder of $$\frac{3^4}{5}$$ is 1

Remainder of $$\frac{3^5}{5}$$ is 3

Here we should recognize the cyclic pattern of remainders. as the power increases remainder continues to move on in a pattern 3421 3421 3421.... so on.

Now if.......

$$7^{11}$$ is completely divisible by 4, then the pattern will stop on 1

if $$7^{11}$$ is divisible by 4 with remainder 1, then the pattern will stop on 3

if $$7^{11}$$ is divisible by 4 with remainder 2, then the pattern will stop on 4

And if $$7^{11}$$ is divisible by 4 with remainder 3, then the pattern will stop on 2

So we basically have to find the remainder when $$7^{11}$$ divided by 4

Rule :- The expression $$\frac{A * B * C}{M}$$ will give the same remainder as $$\frac{Ar * Br * Cr}{M}$$ where Ar, Br, Cr are the remainders of A, B, C when divided by 'M' individually.

$$7^{11}$$ can be simplified as 49*49*49*49*49*7

Remainder of $$\frac{49*49*49*49*49*7}{4}$$ will be the same as that of $$\frac{1*1*1*1*1*7}{4}$$ or that of $$\frac{7}{4}$$

Remainder of $$\frac{7}{4}$$ is 3

Since $$7^{11}$$ divisible by 4 with remainder 3, the pattern will stop on 2 and thus the remainder of$$\frac{3^{7^{11}}}{5}$$ will be 2

Hope that helps!
_________________
Intern
Joined: 02 Jun 2013
Posts: 19
Followers: 0

Kudos [?]: 15 [0], given: 74

Re: What is the remainder of (3^7^11)/5 [#permalink]

### Show Tags

09 Mar 2014, 19:24
hi guys ,

I used wilson reminder therom and found the answer ,

so 3^4/5=1 now we have to make 7^11 in the format of 4x+something so we divide 7^11 by 4 or 3^11 by 4 which will give me reminder of 3 hence 7^11 can be written as 4x+3 ie (3^4x+3)/5 so 3^4x/5 will be 1 3^3 /5 will give reminder 2 ie our answer
Manager
Joined: 10 Mar 2014
Posts: 236
Followers: 1

Kudos [?]: 81 [0], given: 13

Re: PS-What is the remainder [#permalink]

### Show Tags

08 Apr 2014, 03:35
I used the following way

3^7 = 2048. Now (2048)^11. Here last digit is 8 and cycle for this is 8,4,2,6. Now here 11th digit is 2. So when we divide we will get 2 as remainder.
Re: PS-What is the remainder   [#permalink] 08 Apr 2014, 03:35

Go to page    1   2    Next  [ 25 posts ]

Similar topics Replies Last post
Similar
Topics:
1 When n is divided by 24, the remainder is 5. What is the remainder 5 14 Apr 2016, 23:58
If n divided by 7 has a remainder of 2, what is the remainder when 3 5 21 Mar 2016, 06:28
If 2 is the remainder when m is divided by 5, what is the remainder 7 20 Dec 2015, 04:49
9 When n is divided by 5 the remainder is 3. What is the remainder when 9 21 Jul 2015, 02:11
8 What will be the remainder of: 12345678910/8 7 19 Jan 2014, 00:33
Display posts from previous: Sort by