It is currently 21 Oct 2017, 09:14

### 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 when 7^74 - 5^74 is divided by 24?

Author Message
TAGS:

### Hide Tags

Manager
Joined: 04 Sep 2007
Posts: 210

Kudos [?]: 20 [0], given: 0

What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

03 Jul 2008, 13:19
11
This post was
BOOKMARKED
00:00

Difficulty:

45% (medium)

Question Stats:

65% (01:20) correct 35% (01:51) wrong based on 228 sessions

### HideShow timer Statistics

What is the remainder when 7^74 - 5^74 is divided by 24?

A. 0
B. 1
C. 2
D. 3
E. None of these
[Reveal] Spoiler: OA

Kudos [?]: 20 [0], given: 0

Director
Joined: 01 Jan 2008
Posts: 618

Kudos [?]: 198 [1], given: 1

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

03 Jul 2008, 14:20
1
KUDOS
1
This post was
BOOKMARKED
wizardofwashington wrote:
What is the remainder when 7^74-5^74 is divided by 24?

Is there an easy way to solve this?

easiest way for me: 7^74 - 5^74 = (49)^37-25^37 = (24*2+1)^37 - (24+1)^37 -> remainder is 1^37 - 1^37 = 0

Kudos [?]: 198 [1], given: 1

Senior Manager
Joined: 12 Apr 2008
Posts: 499

Kudos [?]: 247 [3], given: 4

Location: Eastern Europe
Schools: Oxford
Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

03 Jul 2008, 14:45
3
KUDOS
5
This post was
BOOKMARKED

It may be useful to remember that a^n-b^n is always divisible by (a-b).

So, when we write 49^37-25^37, we can note that 49-25=24, and thus, the expression can be evenly divided by 24.

Kudos [?]: 247 [3], given: 4

Manager
Joined: 04 Sep 2007
Posts: 210

Kudos [?]: 20 [0], given: 0

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

03 Jul 2008, 14:57
greenoak wrote:

It may be useful to remember that a^n-b^n is always divisible by (a-b).

So, when we write 49^37-25^37, we can note that 49-25=24, and thus, the expression can be evenly divided by 24.

Thanks, guys. Appreciate your quick response.

Kudos [?]: 20 [0], given: 0

SVP
Joined: 29 Aug 2007
Posts: 2472

Kudos [?]: 843 [1], given: 19

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

04 Jul 2008, 23:20
1
KUDOS
maratikus wrote:
wizardofwashington wrote:
What is the remainder when 7^74-5^74 is divided by 24?

Is there an easy way to solve this?

easiest way for me: 7^74 - 5^74 = (49)^37-25^37 = (24*2+1)^37 - (24+1)^37 -> remainder is 1^37 - 1^37 = 0

I like your approach. however i figured out as under:

7^1 - 5^1 = 2 so reminder = 2.
7^2 - 5^2 = 24 ...... so reminder 0.
7^3 - 5^3 = 48 ...... so reminder 0.
7^4 - 5^4 = 218 ....... so reminder = 2

similarly;
7^73 - 5^73 should have a reminder of 2.
7^74 - 5^74 should have a reminder of 0.
_________________

Gmat: http://gmatclub.com/forum/everything-you-need-to-prepare-for-the-gmat-revised-77983.html

GT

Kudos [?]: 843 [1], given: 19

Intern
Joined: 08 Jan 2015
Posts: 13

Kudos [?]: 7 [0], given: 9

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

04 Feb 2015, 05:43
7^54 /24 leaves a remainder of 1. similary 5^54 leaves a remainder of 1 => 1-1=0

Kudos [?]: 7 [0], given: 9

Math Forum Moderator
Joined: 02 Aug 2009
Posts: 4986

Kudos [?]: 5504 [0], given: 112

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

04 Feb 2015, 06:22
Expert's post
5
This post was
BOOKMARKED
abdulfmk wrote:
7^54 /24 leaves a remainder of 1. similary 5^54 leaves a remainder of 1 => 1-1=0

hi abdul..
there is a straight formula for such kind of questions..
1) a^n-b^n is divisible by both a-b and a+b if n is even..
2)a^n-b^n is divisible by a-b if n is odd.
3)a^n+b^n is divisible by a+b if n is odd...
we will use 2 and 3 here..

so 7^54-5^54 = (7^37-5^37)(7^37+5^37)..
now we will use 2 and 3 here..
so (7^37-5^37) is div by 7-5 =2 and (7^37+5^37) by 7+5 or 12..
combined by 2*12 or 24
so remainder 0..
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

Kudos [?]: 5504 [0], given: 112

Manager
Joined: 04 Oct 2013
Posts: 176

Kudos [?]: 157 [0], given: 29

GMAT 1: 590 Q40 V30
GMAT 2: 730 Q49 V40
WE: Project Management (Entertainment and Sports)
What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

04 Feb 2015, 06:25
wizardofwashington wrote:
What is the remainder when 7^74 - 5^74 is divided by 24?

A. 0
B. 1
C. 2
D. 3
E. None of these

If you test a couple of divisions a pattern emerges

7/24 reminder 7; $$7^2$$/24 reminder 1; $$7^3$$/24 reminder 7. We can conclude that when 7^even non-negative integer, reminder is going to be 1; when 7^odd positive integer, reminder is going to be 7.

Thus $$7^7^4$$ will have a reminder of 1

except for 5^1/24, which yields reminder 5. 5^2/24 yields reminder 1, 5^3/24 yields reminder 5, 5^4 yields reminder 1. Thus we can assume that 5^74 will yield reminder 1.

Now R1-R1=R0=multiple of 24.

_________________

learn the rules of the game, then play better than anyone else.

Kudos [?]: 157 [0], given: 29

Math Forum Moderator
Joined: 02 Aug 2009
Posts: 4986

Kudos [?]: 5504 [0], given: 112

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

04 Feb 2015, 07:16
gmat6nplus1 wrote:
wizardofwashington wrote:
What is the remainder when 7^74 - 5^74 is divided by 24?

A. 0
B. 1
C. 2
D. 3
E. None of these

If you test a couple of divisions a pattern emerges

7/24 reminder 7; $$7^2$$/24 reminder 1; $$7^3$$/24 reminder 7. We can conclude that when 7^even non-negative integer, reminder is going to be 1; when 7^odd positive integer, reminder is going to be 7.

Thus $$7^7^4$$ will have a reminder of 1

except for 5^1/24, which yields reminder 5. 5^2/24 yields reminder 1, 5^3/24 yields reminder 5, 5^4 yields reminder 1. Thus we can assume that 5^74 will yield reminder 1.

Now R1-R1=R0=multiple of 24.

hi gmat6nplus1,
there are various ways to do these type of questions ..
but remember, its all about time, so very important to the easiest way .
ill just tell u three ways ..
1) just explained above by me. if u know these rules, the ans will take exactly 10 seconds..
2) as you have written by finding a pattern. may be slightly time consuming.
3) remainder theorem... for example this very Q..
mod for 24or 2^3*3 here will be=2^3*3*(1/2)(2/3)=8. It means for 24, whatever it has to divide say 'a', a^8x will be divisible by 24..
now back to the Q.. 7^74 =7^(8*9+2)= 7^(8*9)+7^2=0 +49..
similarily 5^74=5^(8*9+2)= 5^(8*9)+7^2=0 +25..
combining remainder of 7^74 - 5^74 = 49-25=24, which itself is div by 24...
here this method takes a bit longer but required where the eq is not of this form..
_________________

Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

Kudos [?]: 5504 [0], given: 112

Manager
Joined: 04 Oct 2013
Posts: 176

Kudos [?]: 157 [0], given: 29

GMAT 1: 590 Q40 V30
GMAT 2: 730 Q49 V40
WE: Project Management (Entertainment and Sports)
Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

04 Feb 2015, 07:53
Quote:
hi gmat6nplus1,
there are various ways to do these type of questions ..
but remember, its all about time, so very important to the easiest way .
ill just tell u three ways ..
1) just explained above by me. if u know these rules, the ans will take exactly 10 seconds..
2) as you have written by finding a pattern. may be slightly time consuming.
3) remainder theorem... for example this very Q..
mod for 24or 2^3*3 here will be=2^3*3*(1/2)(2/3)=8. It means for 24, whatever it has to divide say 'a', a^8x will be divisible by 24..
now back to the Q.. 7^74 =7^(8*9+2)= 7^(8*9)+7^2=0 +49..
similarily 5^74=5^(8*9+2)= 5^(8*9)+7^2=0 +25..
combining remainder of 7^74 - 5^74 = 49-25=24, which itself is div by 24...
here this method takes a bit longer but required where the eq is not of this form..

You're right, I was just trying to offer an alternative point of view to this question. Your approach is definitely the fastest
_________________

learn the rules of the game, then play better than anyone else.

Kudos [?]: 157 [0], given: 29

Manager
Status: Gmat Prep
Joined: 22 Jul 2011
Posts: 74

Kudos [?]: 63 [1], given: 39

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

04 Feb 2015, 08:17
1
KUDOS
a^n – b^n = (a – b)(a^n – 1 + a^(n – 2)b + a^(n – 3)b^2 + ··· + + ab^(n – 2) + b^(n – 1))

This might be useful.

Kudos [?]: 63 [1], given: 39

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16598

Kudos [?]: 273 [0], given: 0

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

05 Mar 2016, 05:41
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.
_________________

Kudos [?]: 273 [0], given: 0

BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 2212

Kudos [?]: 843 [0], given: 595

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

17 Mar 2016, 06:03
I actually didi it this way=>
7-5=> remainder =2
7^2-5^2=> remainder =0
7^3-5^3 => remainder =2
hence the pattern => 2,0,2,0.....
hence remainder =>0 as 74 is even
if the terms were both 73 => the remainder would be 2
hence A

Alternatively dont get into this mess => use the identity
hence A^2-B^2=> hence answer => zero
i will try and do it this way next time

peace out
Stone cold
_________________

Give me a hell yeah ...!!!!!

Kudos [?]: 843 [0], given: 595

BSchool Forum Moderator
Joined: 12 Aug 2015
Posts: 2212

Kudos [?]: 843 [0], given: 595

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

21 Mar 2016, 08:58
Okay I don't know many rules as most you do But i surely know binomial.
Here is what i did
7^74 => 49^37 => (48+1)^37 => 12P +1 for some p
and 5^74 => (24+1)^37 => 12Q +1
now subtracting them => 12P+1 - 12Q -1 => 12 (P-Q) => remainder => ZERO
i hope it helps ..
Peace out
_________________

Give me a hell yeah ...!!!!!

Kudos [?]: 843 [0], given: 595

GMAT Club Legend
Joined: 09 Sep 2013
Posts: 16598

Kudos [?]: 273 [0], given: 0

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

21 May 2017, 15:00
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.
_________________

Kudos [?]: 273 [0], given: 0

Intern
Joined: 20 May 2017
Posts: 10

Kudos [?]: 1 [0], given: 2

Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]

### Show Tags

21 May 2017, 15:37
greenoak wrote:

It may be useful to remember that a^n-b^n is always divisible by (a-b).

So, when we write 49^37-25^37, we can note that 49-25=24, and thus, the expression can be evenly divided by 24.

This is by far the cleanest fastest answer.

Kudos [?]: 1 [0], given: 2

Re: What is the remainder when 7^74 - 5^74 is divided by 24?   [#permalink] 21 May 2017, 15:37
Display posts from previous: Sort by