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.

It appears that you are browsing the GMAT Club forum unregistered!

Signing up is free, quick, and confidential.
Join other 500,000 members and get the full benefits of GMAT Club

Registration gives you:

Tests

Take 11 tests and quizzes from GMAT Club and leading GMAT prep companies such as Manhattan GMAT,
Knewton, and others. All are free for GMAT Club members.

Applicant Stats

View detailed applicant stats such as GPA, GMAT score, work experience, location, application
status, and more

Books/Downloads

Download thousands of study notes,
question collections, GMAT Club’s
Grammar and Math books.
All are free!

Thank you for using the timer!
We noticed you are actually not timing your practice. Click the START button first next time you use the timer.
There are many benefits to timing your practice, including:

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..
_________________

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.

Answer A
_________________

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

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.

Answer A

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..
_________________

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.

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.
_________________

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

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
_________________

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.
_________________