GMAT Changed on April 16th - Read about the latest changes here

 It is currently 22 May 2018, 04:56

### 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

# When 51^25 is divided by 13, the remainder obtained is:

Author Message
TAGS:

### Hide Tags

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8066
Location: Pune, India
Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

### Show Tags

11 Jan 2016, 23:39
1
KUDOS
Expert's post
1
This post was
BOOKMARKED
naveenq wrote:
E.g. What is the remainder when 10^5 is divided by 13?

From binomial method, we can take (13-3)^5/13
Next, we're left with (-3)^5/13

Two things - 1st, if I go the long route and take patterns then
3^1 / 13 = R3
3^2 / 13 = R4
3^3 / 13 = R1
and the pattern repeats, in 3's.

so, now, (-3)^5, i know the answer is R4.

However, finding the pattern is time-consuming. Can you help me understand how I can use the binomial theorem for the negative base?
(-3)^5 / 13

THANK YOU!!

(-3)^5/13
Since the power is odd, the negative sign remains. Don't worry about it just yet.
-3^5/13

Note that 3^3 = 27 which is 1 more than a multiple of 13.

-27 * 9/13
-(26+1) * 9/13
Remainder is -9/13

So we have a remainder of -9. This means that remainder is actually 4.

For more on negative remainders, see: http://www.veritasprep.com/blog/2014/03 ... -the-gmat/
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 10 Feb 2011 Posts: 3 When 51^25 is divided by 13, the remainder obtained is: [#permalink] ### Show Tags 04 Dec 2016, 11:49 VeritasPrepKarishma wrote: LM wrote: When 51^25 is divided by 13, the remainder obtained is: A. 12 B. 10 C. 2 D. 1 E. 0 The method used by Bunuel above is the best way to get to the answer. Some time back, I wrote a post detailing the method. Here is the link: (Had to delete the veritas link since I have never posted before on GMAT Club) Once you go through it, this question should be very easy for you. I haven't been able to go through all the replies on this post here and I don't consider myself great at math, but wanted to look at an alternative way of doing this question and remaining within the domains of the GMAT topics (I'd think it would be harsh if the people there started checking us on advanced topics like Binomial Theorem). So let's do prime factorization. 51^25 = (3*17)^25 = (3^25)*(17^25) Using the Last Digit of a Power (pg 14 of GMAT Club Math Book for people unaware of this method), powers of 3 have cyclicality of 4, therefore last digit (3^25) = last digit(3^1) 17^25 would basically check cyclicality of the power of 7 (which is 4), i.e. last digit (17^25) = last digit (17^1). We can therefore simply write (3*17)^25 as (3*17) or 51. When you divide 51/13, you get 12 as the remainder. I can't judge if it is as quicker as the method above, but I definitely think it is simpler to understand. Intern Joined: 22 Jul 2016 Posts: 11 Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink] ### Show Tags 04 Dec 2016, 20:13 I don't get 12 when I plug (51^25)/13 in to a calculator to see what comes out. What am I misunderstanding about the question? Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8066 Location: Pune, India Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink] ### Show Tags 04 Dec 2016, 20:59 Expert's post 1 This post was BOOKMARKED DonnieDrastic wrote: I don't get 12 when I plug (51^25)/13 in to a calculator to see what comes out. What am I misunderstanding about the question? There are two different ways of expressing the result of a division: Say, I tell you the following: Divide 11 by 4. What do you get? You could answer me with one of the following: Case 1: You could say, “I get 2.75” Case 2: You could say, “I get 2 as the quotient and 3 as the remainder.” Either ways, you are correct. 11/4 = (2 ¾) When you use the decimal form, you get a .75 which you add to 2 to give you 2.75. This .75 is nothing but the way you express the remainder 3. When you divide 11 by 4, 4 goes into 11 two times and then 3 is left over. When 4 goes into 3, you get 0.75 which is ¾. That is the reason why you can write 11/4 as (2 ¾) in mixed fractions. The calculator gives you the result of case 1. Anyway, the calculator will give you an approximate value. This question asks you for the remainder, as in case 2. _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8066
Location: Pune, India
Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

### Show Tags

04 Dec 2016, 22:05
omairanw wrote:
VeritasPrepKarishma wrote:
LM wrote:
When 51^25 is divided by 13, the remainder obtained is:

A. 12
B. 10
C. 2
D. 1
E. 0

The method used by Bunuel above is the best way to get to the answer. Some time back, I wrote a post detailing the method. Here is the link:

(Had to delete the veritas link since I have never posted before on GMAT Club)

Once you go through it, this question should be very easy for you.

I haven't been able to go through all the replies on this post here and I don't consider myself great at math, but wanted to look at an alternative way of doing this question and remaining within the domains of the GMAT topics (I'd think it would be harsh if the people there started checking us on advanced topics like Binomial Theorem).

So let's do prime factorization. 51^25 = (3*17)^25 = (3^25)*(17^25)

Using the Last Digit of a Power (pg 14 of GMAT Club Math Book for people unaware of this method), powers of 3 have cyclicality of 4, therefore last digit (3^25) = last digit(3^1)
17^25 would basically check cyclicality of the power of 7 (which is 4), i.e. last digit (17^25) = last digit (17^1). We can therefore simply write (3*17)^25 as (3*17) or 51. When you divide 51/13, you get 12 as the remainder. I can't judge if it is as quicker as the method above, but I definitely think it is simpler to understand.

Cyclicity can help you find the remainder only in case of division by 2, 5 or 10. Check this post to know why:
https://www.veritasprep.com/blog/2015/1 ... questions/

The last digit will not decide the remainder in every case in case the divisor is other than 2, 5 or 10.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199 Veritas Prep Reviews Intern Joined: 02 Oct 2016 Posts: 35 Schools: HEC Dec '17 Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink] ### Show Tags 24 Jan 2017, 01:59 Bunuel Hi Bunuel, I solved this using the remainder and cyclicity method and got the correct answer. Can you please tell me if this is just a coincidence? By applying cyclicity of 4: 51^25 ~ 51^1 Therefore, 51/13 ---> Remainder = 12 Math Expert Joined: 02 Sep 2009 Posts: 45245 Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink] ### Show Tags 24 Jan 2017, 02:42 Vaidya wrote: Bunuel Hi Bunuel, I solved this using the remainder and cyclicity method and got the correct answer. Can you please tell me if this is just a coincidence? By applying cyclicity of 4: 51^25 ~ 51^1 Therefore, 51/13 ---> Remainder = 12 The question is how do you know the cycle? In fact the remainder repeats in blocks of two here: The remainder of 51^1 divided by 13 is 12; The remainder of 51^2 divided by 13 is 1; The remainder of 51^3 divided by 13 is 12; The remainder of 51^4 divided by 13 is 1; The remainder of 51^5 divided by 13 is 12; ... The remainder of 51^odd divided by 13 is 12; The remainder of 51^even divided by 13 is 1. _________________ Senior Manager Status: love the club... Joined: 24 Mar 2015 Posts: 274 Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink] ### Show Tags 04 Sep 2017, 00:33 Bunuel wrote: LM wrote: When 51^25 is divided by 13, the remainder obtained is: A. 12 B. 10 C. 2 D. 1 E. 0 $$51^{25}=(52-1)^{25}$$, now if we expand this expression all terms but the last one will have $$52=13*4$$ in them, thus will leave no remainder upon division by 13, the last term will be $$(-1)^{25}=-1$$. Thus the question becomes: what is the remainder upon division -1 by 13? The answer to this question is 12: $$-1=13*(-1)+12$$. Answer: A. hi 51^25 here the last digit of 51 is 1, and the cyclicity of 1 is 1. So, the unit digit of 51^25 will be 1 thus, 1, when divided by 13 will produce a quotient "0" and will leave a remainder "1", and I am stumped here ... please help me understand the problem ... thanks in advance ... Math Expert Joined: 02 Sep 2009 Posts: 45245 Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink] ### Show Tags 04 Sep 2017, 00:36 gmatcracker2017 wrote: Bunuel wrote: LM wrote: When 51^25 is divided by 13, the remainder obtained is: A. 12 B. 10 C. 2 D. 1 E. 0 $$51^{25}=(52-1)^{25}$$, now if we expand this expression all terms but the last one will have $$52=13*4$$ in them, thus will leave no remainder upon division by 13, the last term will be $$(-1)^{25}=-1$$. Thus the question becomes: what is the remainder upon division -1 by 13? The answer to this question is 12: $$-1=13*(-1)+12$$. Answer: A. hi 51^25 here the last digit of 51 is 1, and the cyclicity of 1 is 1. So, the unit digit of 51^25 will be 1 thus, 1, when divided by 13 will produce a quotient "0" and will leave a remainder "1", and I am stumped here ... please help me understand the problem ... thanks in advance ... You cannot do this way. For example, 1 divided by 13 gives the remainder of 1 but 21 divided by 13 gives the remainder of 8. There are many different approaches and links to the underlying theory on previous 5 pages of the discussion. _________________ Manager Joined: 24 Jun 2017 Posts: 122 Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink] ### Show Tags 04 Sep 2017, 16:33 via fermat theorem a^(p-1)= 1 mod p where a is an integer and p prime as 51 contains 13 it can be rewritten and simplified 51/13 remainder 12 12^(13-1) = 1 mod 13 12^25 = 12^(12*2+1)= 12 Manager Joined: 21 Feb 2017 Posts: 81 Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink] ### Show Tags 29 Sep 2017, 00:13 [i] Hello Bunuel and Karishma, Can anyone from you, please tell me the limitation of cyclic rule? So far i was using that rule blindly for all question. But it has not worked for this particular problem. [i] Veritas Prep GMAT Instructor Joined: 16 Oct 2010 Posts: 8066 Location: Pune, India Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink] ### Show Tags 29 Sep 2017, 01:12 goalMBA1990 wrote: [i] Hello Bunuel and Karishma, Can anyone from you, please tell me the limitation of cyclic rule? So far i was using that rule blindly for all question. But it has not worked for this particular problem. [i] Cyclicity helps you figure out the units digit in every case. Also, it has nothing to do with remainders. In some cases, the units digit helps you figure out the remainder, more specifically in case of division by 2, 5 or 10. https://www.veritasprep.com/blog/2015/1 ... questions/ https://www.veritasprep.com/blog/2015/1 ... ns-part-2/ _________________ Karishma Veritas Prep | GMAT Instructor My Blog Get started with Veritas Prep GMAT On Demand for$199

Veritas Prep Reviews

Intern
Joined: 02 Oct 2016
Posts: 6
Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

### Show Tags

21 Feb 2018, 08:07
LM wrote:
When 51^25 is divided by 13, the remainder obtained is:

A. 12
B. 10
C. 2
D. 1
E. 0

Remainder (A * B) = R(A)*R(B)

Therefore 51^25/13 = [51*51*51*51....(25 times)] / 13
= [12*12*12.........(25 times)] / 13 (As 51/13 is 12)

= [144*144..........(12 times) *12] / 13 ( Grouping two adjacent 12's together so finally one 12 will be left)
= [1*1*1.............(12 times) *12] / 13 (144/13 is 1)
= 12 / 13
= 12
Director
Status: It's near - I can see.
Joined: 13 Apr 2013
Posts: 946
Location: India
GMAT 1: 480 Q38 V22
GPA: 3.01
WE: Engineering (Consulting)
Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

### Show Tags

29 Mar 2018, 23:02
VeritasPrepKarishma wrote:
LM wrote:
When 51^25 is divided by 13, the remainder obtained is:

A. 12
B. 10
C. 2
D. 1
E. 0

The method used by Bunuel above is the best way to get to the answer. Some time back, I wrote a post detailing the method. Here is the link:

http://www.veritasprep.com/blog/2011/05 ... ek-in-you/

Once you go through it, this question should be very easy for you.

Mam, why cyclicity method does not work here. I marked answer 1 thinking 1/13 remainder "1"
_________________

"Success is not as glamorous as people tell you. It's a lot of hours spent in the darkness."

Senior Manager
Status: love the club...
Joined: 24 Mar 2015
Posts: 274
Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

### Show Tags

30 Mar 2018, 02:09
disharupani wrote:
Hey, I didn't understand:

When -1 is divided by 13 how can the reminder be 12?
If I substitute in the equation -1=13q+r, so how can we arrive at r=12?

hi

okay, think about simple division, when you divide 5 by 2, you get 2 as quotient and 1 as remainder, because 4 is the greatest integer less than 5

now, on the same line of reasoning, when you divide -1 by 13, -13 is the greatest integer less than -1, and thus 12 is the remainder

hope this helps and is clear!
thanks and cheers!
Veritas Prep GMAT Instructor
Joined: 16 Oct 2010
Posts: 8066
Location: Pune, India
Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

### Show Tags

30 Mar 2018, 02:26
QZ wrote:
VeritasPrepKarishma wrote:
LM wrote:
When 51^25 is divided by 13, the remainder obtained is:

A. 12
B. 10
C. 2
D. 1
E. 0

The method used by Bunuel above is the best way to get to the answer. Some time back, I wrote a post detailing the method. Here is the link:

http://www.veritasprep.com/blog/2011/05 ... ek-in-you/

Once you go through it, this question should be very easy for you.

Mam, why cyclicity method does not work here. I marked answer 1 thinking 1/13 remainder "1"

You are not dividing 1 by 13 here. You are dividing a number that ends in 1 by 13 here. We don't know what that number exactly is.
Will every number ending in 1 when divided by 13 give 1 as remainder? Think about it:
11/13, remainder 11
21/13, remainder 8
31/13, remainder 5
and so on...
Cyclicity does not help get the remainder in most cases. It does so in very few cases. They are discussed in this post: https://www.veritasprep.com/blog/2015/1 ... questions/
https://www.veritasprep.com/blog/2015/1 ... ns-part-2/
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for \$199

Veritas Prep Reviews

Intern
Joined: 02 Oct 2016
Posts: 25
Re: When 51^25 is divided by 13, the remainder obtained is: [#permalink]

### Show Tags

08 Apr 2018, 06:58
1
KUDOS
5125=(52−1)255125=(52−1)25 , now if we expand this expression all terms but the last one will have 52=13∗452=13∗4 in them, thus will leave no remainder upon division by 13, the last term will be (−1)25=−1(−1)25=−1. Thus the question becomes: what is the remainder upon division -1 by 13? The answer to this question is 12: −1=13∗(−1)+12−1=13∗(−1)+12.

Re: When 51^25 is divided by 13, the remainder obtained is:   [#permalink] 08 Apr 2018, 06:58

Go to page   Previous    1   2   3   [ 57 posts ]

Display posts from previous: Sort by