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What is the remainder when (3^84)/26

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ahmed.abumera
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Re: What is the remainder when (3^84)/26 [#permalink] New post   04 Feb 2018, 02:18
Dear Experts ,

Please let me know what is the problem in the method below;

3^84 = 9^42 = 81^21. so the question would be what is the remainder when (78+3)^21 divided by 26. since 26 is a factor of 78 that yields no remainder when dividing by 26. so we have to find the remainder when 3^21 divided by 26. Cyclisty tells us 3^21 ends with 3 in the units . so the remainder when 3 is divided by 26 is 3..
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Re: What is the remainder when (3^84)/26 [#permalink] New post   02 Sep 2018, 18:49
Abhishek009 wrote:
Bunuel wrote:
What is the remainder when \(\frac{(3^{84})}{26}\)

(A) 0
(B) 1
(C) 2
(D) 24
(E) 25

\(\frac{(3^{84})}{26}\)

= \(\frac{3^{3*28}}{26}\)

= \(\frac{27^{28}}{26}\)

= \(\frac{(26 + 1)^{28}}{26}\)

= \(\frac{(26^{28} + 1^{28})}{26}\)

\frac{26^{Any \ Number }}{26} , will have remainder 0
\frac{1^{Any \ Number }}{26} , will have remainder 1

So, The answer will be (B) 1




Hi Abhishek009,

I am slightly confused = is it not wrong to say that 27^28 = (26^28 + 1^28) ??

it's like saying 2^4 = (1^4 + 1^4) which comes out to be = 2 and not 16..

Please guide?
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CEdward
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Re: What is the remainder when (3^84)/26 [#permalink] New post   02 Mar 2021, 22:32
Would this same method of cyclical remainders be applicable?

3^1 = 3 --> 3/26 --> R3
3^2 = 9 --> 9/26 --> R9
3^3 = 27 --> 27/26 --> 1 R1 <---- Cycle stops here
3^4 = 81 --> 81 / 26 --> 3 R3

Given an exponent of 84 ---> 84 /3 = 26 <--- so 26 cycles of 3. Hence, the remainder is 1.
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Nik11
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What is the remainder when (3^84)/26 [#permalink] New post   13 Mar 2021, 23:42
Nik11 wrote:
ashikaverma13 wrote:
Bunuel wrote:
What is the remainder when \(\frac{(3^{84})}{26}\)

(A) 0
(B) 1
(C) 2
(D) 24
(E) 25


Hi Bunuel,

Will it be possible for you to provide an approach to this type of questions? or could you guide me to a post where you have already posted an explanation to such questions? I read the entire thread and I was unable to understand the approaches and frankly it seemed like a really time consuming approaches.
Kindly help,

thanks in advance!


For this type of questions for me the best approach is to use cycles. Even though sometimes could take you instead of 50 sec 1:50. This simply because is the only one among the 3 that you can apply in "many" others mid-high level problems, but just my tought.

3^1 when divided by has remainder equal to 26
3^2 9
3^3 1
3^4 3 probably cycle yet finished
3^5 9 I would be sufficiently sure and I wouldn't go ahead but your choice
so the pattern is 9, 1, 3
84 is divisible by 3 hence r = 3

I want to ask to the experts what is the binomial method and if it worth to study it
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Re: What is the remainder when (3^84)/26 [#permalink] New post   24 May 2023, 18:58
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Re: What is the remainder when (3^84)/26 [#permalink]
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