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What is the remainder when 7^74  5^74 is divided by 24?
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03 Jul 2008, 13:19
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67% (01:50) correct 33% (02:02) wrong based on 360 sessions
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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
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Re: What is the remainder when 7^74  5^74 is divided by 24?
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04 Feb 2015, 06:22
abdulfmk wrote: 7^54 /24 leaves a remainder of 1. similary 5^54 leaves a remainder of 1 => 11=0 hi abdul.. there is a straight formula for such kind of questions.. 1) a^nb^n is divisible by both ab and a+b if n is even.. 2)a^nb^n is divisible by ab 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 7^545^54 = (7^375^37)(7^37+5^37).. now we will use 2 and 3 here.. so (7^375^37) is div by 75 =2 and (7^37+5^37) by 7+5 or 12.. combined by 2*12 or 24 so remainder 0..
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Re: What is the remainder when 7^74  5^74 is divided by 24?
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03 Jul 2008, 14:45
In addition to maraticus's solution:
It may be useful to remember that a^nb^n is always divisible by (ab).
So, when we write 49^3725^37, we can note that 4925=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?
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03 Jul 2008, 14:20
wizardofwashington wrote: What is the remainder when 7^745^74 is divided by 24?
Is there an easy way to solve this? easiest way for me: 7^74  5^74 = (49)^3725^37 = (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?
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03 Jul 2008, 14:57
greenoak wrote: In addition to maraticus's solution:
It may be useful to remember that a^nb^n is always divisible by (ab).
So, when we write 49^3725^37, we can note that 4925=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?
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04 Jul 2008, 23:20
maratikus wrote: wizardofwashington wrote: What is the remainder when 7^745^74 is divided by 24?
Is there an easy way to solve this? easiest way for me: 7^74  5^74 = (49)^3725^37 = (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?
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04 Feb 2015, 05:43
7^54 /24 leaves a remainder of 1. similary 5^54 leaves a remainder of 1 => 11=0



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What is the remainder when 7^74  5^74 is divided by 24?
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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 nonnegative 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 R1R1=R0=multiple of 24. Answer A
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Re: What is the remainder when 7^74  5^74 is divided by 24?
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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 nonnegative 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 R1R1=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 = 4925=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?
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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 = 4925=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?
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04 Feb 2015, 08:17
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.



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Re: What is the remainder when 7^74  5^74 is divided by 24?
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17 Mar 2016, 06:03
I actually didi it this way=> 75=> remainder =2 7^25^2=> remainder =0 7^35^3 => 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^2B^2=> hence answer => zero i will try and do it this way next time peace out Stone cold
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Re: What is the remainder when 7^74  5^74 is divided by 24?
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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 (PQ) => remainder => ZERO i hope it helps .. Peace out
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Re: What is the remainder when 7^74  5^74 is divided by 24?
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21 May 2017, 15:37
greenoak wrote: In addition to maraticus's solution:
It may be useful to remember that a^nb^n is always divisible by (ab).
So, when we write 49^3725^37, we can note that 4925=24, and thus, the expression can be evenly divided by 24. This is by far the cleanest fastest answer.



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Re: What is the remainder when 7^74  5^74 is divided by 24?
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16 Jun 2018, 09:56
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 Note that 7^2 = 49 produces a remainder of 1 when divided by 24. Since 7^3 = 7^2 x 7, the remainder from the division of 7^3 by 24 is 7 and since 7^4 = 7^2 x 7^2, the remainder from the division of 7^4 by 24 is 1. We can generalize this as follows: 7^n produces a remainder of 1 when n is even and produces a remainder of 7 when n is odd. Similarly, 5^2 = 25 produces a remainder of 1 when divided by 24. Since 5^3 = 5^2 x 5, the remainder from the division of 5^3 by 24 is 5 and since 5^4 = 5^2 x 5^2, the remainder from the division of 5^4 by 24 is 1. We can generalize this as follows: 5^n produces a remainder of 1 when n is even and produces a remainder of 5 when n is odd. Thus, the remainder from division of 7^74 and 5^74 by 24 are both 1 and the remainder from the division of 7^74  5^74 is 1  1 = 0. Answer: A
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Re: What is the remainder when 7^74  5^74 is divided by 24?
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05 Aug 2018, 20:45
Hi All,
Will it be possible to find the answer using the cyclicity of digits?
Thanks



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Re: What is the remainder when 7^74  5^74 is divided by 24?
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05 Jan 2019, 13:01
chetan2u wrote: abdulfmk wrote: 7^54 /24 leaves a remainder of 1. similary 5^54 leaves a remainder of 1 => 11=0 hi abdul.. there is a straight formula for such kind of questions.. 1) a^nb^n is divisible by both ab and a+b if n is even.. 2)a^nb^n is divisible by ab 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 7^545^54 = (7^375^37)(7^37+5^37).. now we will use 2 and 3 here.. so (7^375^37) is div by 75 =2 and (7^37+5^37) by 7+5 or 12.. combined by 2*12 or 24 so remainder 0.. Hello! Could you please explain why are we using rules 2 and 3 if in this particular problem n=74=enven? Kind regards!




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