What is the remainder when (3^84)/26 : Problem Solving (PS)

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

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Intern

Joined: 27 Sep 2017

Posts: 10

Re: What is the remainder when (3^84)/26 [#permalink]

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04 Feb 2018, 01:18

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..

Intern

Joined: 30 Apr 2018

Posts: 19

Re: What is the remainder when (3^84)/26 [#permalink]

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02 Sep 2018, 17:49

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?