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

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Intern
Joined: 27 Sep 2017
Posts: 9
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: 17
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..

Intern
Joined: 02 Jul 2019
Posts: 1
Re: What is the remainder when (3^84)/26  [#permalink]

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18 Sep 2019, 11:30
NidSha wrote:
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..

Posted from my mobile device
Re: What is the remainder when (3^84)/26   [#permalink] 18 Sep 2019, 11:30

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