January 17, 2019 January 17, 2019 08:00 AM PST 09:00 AM PST Learn the winning strategy for a high GRE score — what do people who reach a high score do differently? We're going to share insights, tips and strategies from data we've collected from over 50,000 students who used examPAL. January 19, 2019 January 19, 2019 07:00 AM PST 09:00 AM PST Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.
Author 
Message 
Director
Joined: 03 Sep 2006
Posts: 785

What is the remainder of (3^7^11)/5
[#permalink]
Show Tags
24 Jan 2012, 07:22
Question Stats:
50% (01:04) correct 50% (00:59) wrong based on 1036 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?
Official Answer and Stats are available only to registered users. Register/ Login.



Manager
Status: Employed
Joined: 17 Nov 2011
Posts: 80
Location: Pakistan
Concentration: International Business, Marketing
GPA: 3.2
WE: Business Development (Internet and New Media)

Re: PSWhat is the remainder
[#permalink]
Show Tags
24 Jan 2012, 07:43
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
GPA: 3.29
WE: Engineering (Other)

Re: PSWhat is the remainder
[#permalink]
Show Tags
24 Jan 2012, 11:31
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 useUnderstand 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) \(3^5=24 3\)  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 \((81)^{11}\) Divide \((81)^{11}\) 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 remainderHence \(7^{11}\) 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: 52210

What is the remainder of (3^7^11)/5
[#permalink]
Show Tags
24 Jan 2012, 12:15
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 3 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. Answer: C. 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: \(\frac{3^{7^{11}}}{5}\)
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8787
Location: Pune, India

Re: PSWhat is the remainder
[#permalink]
Show Tags
26 Jan 2012, 00:41
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 ... yapplied/http://www.veritasprep.com/blog/2011/05 ... emainders/
_________________
Karishma Veritas Prep GMAT Instructor
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Math Expert
Joined: 02 Sep 2009
Posts: 52210

Re: PSWhat is the remainder
[#permalink]
Show Tags
26 Jan 2012, 03:52



Intern
Joined: 23 Jan 2012
Posts: 7
Location: India
Concentration: Strategy, Finance
GPA: 3.29
WE: Engineering (Other)

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: 8787
Location: Pune, India

Re: What is the remainder of (3^7^11)/5
[#permalink]
Show Tags
26 Jan 2012, 21:11
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
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Math Expert
Joined: 02 Sep 2009
Posts: 52210

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.
_________________
New to the Math Forum? Please read this: Ultimate GMAT Quantitative Megathread  All You Need for Quant  PLEASE READ AND FOLLOW: 12 Rules for Posting!!! Resources: GMAT Math Book  Triangles  Polygons  Coordinate Geometry  Factorials  Circles  Number Theory  Remainders; 8. Overlapping Sets  PDF of Math Book; 10. Remainders  GMAT Prep Software Analysis  SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS)  Tricky questions from previous years.
Collection of Questions: PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.
What are GMAT Club Tests? Extrahard Quant Tests with Brilliant Analytics



Intern
Joined: 12 Mar 2012
Posts: 10

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: 8787
Location: Pune, India

Re: What is the remainder of (3^7^11)/5
[#permalink]
Show Tags
25 Mar 2012, 20:22
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. \(x = 7^{11} = (81)^{11}\) so remainder will be 1 i.e. 3 (for explanation of this, check out: http://www.veritasprep.com/blog/2011/05 ... ekinyou/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
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Intern
Joined: 22 Jan 2012
Posts: 20

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: 61

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(81)^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: 160
Location: India
Concentration: Technology, Leadership
GMAT Date: 08022013
GPA: 3.7
WE: Consulting (Consulting)

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: 132
Location: India
Concentration: Finance, Strategy
GMAT 1: 530 Q39 V23 GMAT 2: 560 Q42 V26
GPA: 3.5
WE: Consulting (Computer Hardware)

Re: PSWhat 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 ... yapplied/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 (52)^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: 8787
Location: Pune, India

Re: PSWhat 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 ... yapplied/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 (52)^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
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >



Math Expert
Joined: 02 Sep 2009
Posts: 52210

Re: PSWhat is the remainder
[#permalink]
Show Tags
24 Sep 2013, 06:30



MBA Section Director
Affiliations: GMAT Club
Joined: 21 Feb 2012
Posts: 5989
City: Pune

Re: PSWhat is the remainder
[#permalink]
Show Tags
24 Sep 2013, 13:10
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
(52)^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 4Rule : 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!
_________________
Chances of Getting Admitted After an Interview [Data Crunch]
Must Read Forum Topics Before You Kick Off Your MBA Application
New GMAT Club Decision Tracker  Real Time Decision Updates



Intern
Joined: 02 Jun 2013
Posts: 19

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: 191

Re: PSWhat 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: PSWhat is the remainder &nbs
[#permalink]
08 Apr 2014, 03:35



Go to page
1 2
Next
[ 26 posts ]



