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

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Manager
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What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  03 Jul 2008, 12:19
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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
Director
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Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  03 Jul 2008, 13:20
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
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Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  03 Jul 2008, 13:45
1
KUDOS

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.
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Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  03 Jul 2008, 13:57
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.
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Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  04 Jul 2008, 22:20
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.
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Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  04 Feb 2015, 04:43
7^54 /24 leaves a remainder of 1. similary 5^54 leaves a remainder of 1 => 1-1=0
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Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  04 Feb 2015, 05:22
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..
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What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  04 Feb 2015, 05:25
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.

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Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  04 Feb 2015, 06:16
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..
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Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  04 Feb 2015, 06:53
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
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Re: What is the remainder when 7^74 - 5^74 is divided by 24? [#permalink]  04 Feb 2015, 07:17
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.
Re: What is the remainder when 7^74 - 5^74 is divided by 24?   [#permalink] 04 Feb 2015, 07:17
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