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# What will be the remainder when 13^36 is divided by 2196?

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Intern
Joined: 01 Nov 2015
Posts: 21
What will be the remainder when 13^36 is divided by 2196? [#permalink]

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08 Jun 2016, 18:39
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Difficulty:

65% (hard)

Question Stats:

61% (01:46) correct 39% (02:19) wrong based on 154 sessions

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What will be the remainder when 13^36 is divided by 2196?

A) 0
B) 1
C) 12
D) 2195
E) 5
[Reveal] Spoiler: OA
Math Expert
Joined: 02 Aug 2009
Posts: 5654
Re: What will be the remainder when 13^36 is divided by 2196? [#permalink]

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08 Jun 2016, 18:49
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aayushagrawal wrote:
What will be the remainder when 13^36 is divided by 2196?

A) 0
B) 1
C)12
D) 2195
E) 5

"Please hit +kudos if you like this post"

In such Qs, best is to get the dividend and divisor in some friendly figures..
$$13^{36} = (13^3)^{12} = 2197^{12} = (2196+1)^{12}$$.....
when $$(2196+1)^{12}$$ is divided by 2196, all terms in the expansion are divisible by 2196 except 1^12, so the remainder will be 1
B
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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Re: What will be the remainder when 13^36 is divided by 2196? [#permalink]

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28 Jul 2017, 17:05
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Manager
Joined: 03 May 2014
Posts: 166
Location: India
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Re: What will be the remainder when 13^36 is divided by 2196? [#permalink]

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20 Aug 2017, 19:28
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We can also use cyclicity of the last digit to answer this question
13ˆ36/2196
The remainder will depend on the last digit of Numerator ie on 3.
Cyclicity of 3 is 4 and the last digits are
3¹=3
3²=9
3³=27
3⁴=81
Now 36/4=9 the digit will have 9 full cycles and the end of the cycle has 1 as the last digit
1³⁶/2196 the reminder will be 1
Manager
Joined: 24 Jun 2017
Posts: 121
Re: What will be the remainder when 13^36 is divided by 2196? [#permalink]

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04 Sep 2017, 16:38
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it can be resolved on the fly via cyclicity approach by creating a pattern:
Rem[13^1 / 2196] = Rem [13 / 2196] =13
Rem[13^2 / 2196] = Rem [169 / 2196] =49
Rem[13^3 / 2196] = Rem [2197/ 2196] = 1
Rem[13^4 / 2196] = Rem [2197*13 / 2196] = 13

So the cycle is 3 as on the step four it started repeating the same remainder (you can ignore the 4th line for further calculations)

13^36 = 36 / 3 = 12 so it the last 3rd cycle which is 1
Re: What will be the remainder when 13^36 is divided by 2196?   [#permalink] 04 Sep 2017, 16:38
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