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What is the remainder when (3^84)/26
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ahmed.abumera
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Joined: 27 Sep 2017
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Re: What is the remainder when (3^84)/26 [#permalink] New post  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..
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NidSha
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Re: What is the remainder when (3^84)/26 [#permalink] New post  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?
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cm291093
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Joined: 02 Jul 2019
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Re: What is the remainder when (3^84)/26 [#permalink] New post  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..

Please guide?


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AliciaSierra
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Joined: 17 Mar 2014
Posts: 677
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Re: What is the remainder when (3^84)/26 [#permalink] New post  25 Jul 2020, 09:19
Can I find out remainder using cyclicity?

As per mathematical cyclicity, 3 has cyclicity of 4 i.e. after every 4 terms, unit digit of 3 repeats.
unit digit of
\(3^0= 1\)
\(3^1= 3\)
\(3^2= 9\)
\(3^3= 7\)
\(3^4=1\)
\(3^5=3\)
\(3^6=9\)
\(3^7=7\)

Unit digit of \(3^{84}\)
= unit digit of \(3^{4*21}\)=Unit digit of\( (3^4)^{21}= 1^{21}=1.\)

so unit digit of 3^84/26= 1/26

remainder of 1/26= 1.
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