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What is the remainder when 11^452 is divided by 6

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What is the remainder when 11^452 is divided by 6 [#permalink]

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New post 16 Jun 2017, 00:55
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What is the remainder when \(11^{452}\) is divided by 6.

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
[Reveal] Spoiler: OA

Last edited by Bunuel on 16 Jun 2017, 01:35, edited 1 time in total.
Renamed the topic.

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Re: What is the remainder when 11^452 is divided by 6 [#permalink]

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New post 16 Jun 2017, 00:58
sriamlan wrote:
What is the remainder when \(11^{452}\) is divided by 6.

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5


Experts my doubt here is -
The cyclicity of power of 11 when divided by 6 is 5,4,3,2,1,0

So if we divide 452 by 6 remainder is 2

So as 2 is the remainder as per the cyclicity should not 4 be the answer instead of 1?

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Re: What is the remainder when 11^452 is divided by 6 [#permalink]

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New post 16 Jun 2017, 01:03
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sriamlan wrote:
What is the remainder when \(11^{452}\) is divided by 6.

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5


11 divided by 6 gives remainder 5.

\(11^2 = 121\) divided by 6 gives remainder 1.

\(11^3 = 1331\) divided by 6 gives remainder 5.

\(11^4 = 14641\) divided by 6 gives remainder 1.

Therefore it has a cycle of 2 with 5 and 1 as remainders.

Even powers of 11 divided by 6 gives remainder 1. And Odd powers of 11 divided by 6 gives remainder 5.

452 divided by 2 = 226.

Therefore \(11^{452}\) divided by 6 will give remainder 1.

Answer (A)...

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Re: What is the remainder when 11^452 is divided by 6 [#permalink]

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New post 16 Jun 2017, 01:45
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sriamlan wrote:
What is the remainder when \(11^{452}\) is divided by 6.

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5


\(\frac{11}{6}=\frac{5}{6} \implies \frac{11^{452}}{6}=\frac{5^{452}}{6}=\frac{(-1)^{452}}{6}=\frac{1}{6}\)

The answer is A.
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What is the remainder when 11^452 is divided by 6 [#permalink]

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New post 16 Jun 2017, 09:23
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sriamlan wrote:
What is the remainder when \(11^{452}\) is divided by 6.

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5


\(\frac{11}{6}\) = Remainder \(5\)
\(\frac{11^2}{6}\) = Remainder \(1\)

Now, \(11^{452}\) = \(11^{2*226}\)

\(\frac{11^{2*226}}{6}\) will leave a remainder 1

Thus, the correct answer will be (A) 1
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Re: What is the remainder when 11^452 is divided by 6 [#permalink]

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New post 05 Jul 2017, 11:45
sriamlan wrote:
sriamlan wrote:
What is the remainder when \(11^{452}\) is divided by 6.

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5


Experts my doubt here is -
The cyclicity of power of 11 when divided by 6 is 5,4,3,2,1,0

So if we divide 452 by 6 remainder is 2

So as 2 is the remainder as per the cyclicity should not 4 be the answer instead of 1?



Not sure if i can clear your doubt. You can look at the problem like when 11/6 then remainder is 5, when 11^2/6 then remainder is 1.
Now you have 11^452 which is 11^even power so remainder will be 1.

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Re: What is the remainder when 11^452 is divided by 6 [#permalink]

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New post 04 Sep 2017, 17:08
6 = 3 * 2
11^452 = (11^256)*(11^256)
a^(p-1)/p = 1 mod p

11^(3-1)/3 = 11^2 = 1 mod 3
11^256= 11^(2*128) = 1 mod 3
(11^256)/2 = 1 mod 2

together 1

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Re: What is the remainder when 11^452 is divided by 6 [#permalink]

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New post 19 Sep 2017, 04:36
Rem|11^452/6| is equivalent to Rem|5^452/6|

Now rem|5^2/6| = 1

So Rem|5^452/6| = Rem|(5^2)^226/6|

=> 1^226 = 1

Final Remainder = 1

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Re: What is the remainder when 11^452 is divided by 6 [#permalink]

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New post 28 Sep 2017, 14:33
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sriamlan wrote:
What is the remainder when \(11^{452}\) is divided by 6.

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5


the easiest way ever

(12 - 1)^452
___________
6
now as, 12/6 leaves no remainder
(-1)^ 452
________
6

=1/6
leaves remainder 1

cheers through the kudos button if this helps

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Re: What is the remainder when 11^452 is divided by 6   [#permalink] 28 Sep 2017, 14:33
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