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

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04 Feb 2015, 06:22

Expert's post

2

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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.. _________________

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

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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.

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

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04 Feb 2015, 07:16

Expert's post

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.

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]

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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]

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05 Mar 2016, 05:41

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

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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]

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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

gmatclubot

Re: What is the remainder when 7^74 - 5^74 is divided by 24?
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21 Mar 2016, 08:58

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