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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: 20
What will be the remainder when 13^36 is divided by 2196? [#permalink]

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08 Jun 2016, 19:39
1
4
00:00

Difficulty:

55% (hard)

Question Stats:

61% (01:42) correct 39% (02:10) wrong based on 182 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
Math Expert
Joined: 02 Aug 2009
Posts: 5938
Re: What will be the remainder when 13^36 is divided by 2196? [#permalink]

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08 Jun 2016, 19:49
3
9
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
_________________

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

GMAT online Tutor

Manager
Joined: 04 May 2014
Posts: 162
Location: India
WE: Sales (Mutual Funds and Brokerage)
Re: What will be the remainder when 13^36 is divided by 2196? [#permalink]

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20 Aug 2017, 20:28
1
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: 122
Re: What will be the remainder when 13^36 is divided by 2196? [#permalink]

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04 Sep 2017, 17:38
1
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
Intern
Joined: 04 May 2018
Posts: 3
Re: What will be the remainder when 13^36 is divided by 2196? [#permalink]

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16 May 2018, 01:19
chetan2u wrote:
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

HI Chetan,

why can't we use Cyclicity in here? Is there a particular criteria that has to be fulfilled to use Cyclicity?

TIA
Math Expert
Joined: 02 Aug 2009
Posts: 5938
Re: What will be the remainder when 13^36 is divided by 2196? [#permalink]

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18 May 2018, 00:03
ajtmatch wrote:
chetan2u wrote:
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

HI Chetan,

why can't we use Cyclicity in here? Is there a particular criteria that has to be fulfilled to use Cyclicity?

TIA

hi..
cyclicity generally helps us in knowing the LAST or unit's digit...
so if we are looking for remainder when divided by 10, you can use cyclicity..
_________________

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

GMAT online Tutor

Re: What will be the remainder when 13^36 is divided by 2196?   [#permalink] 18 May 2018, 00:03
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