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

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aayushagrawal
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What will be the remainder when 13^36 is divided by 2196? [#permalink] New post   08 Jun 2016, 19:39
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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
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chetan2u
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Re: What will be the remainder when 13^36 is divided by 2196? [#permalink] New post   08 Jun 2016, 19:49
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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" :-D



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
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gps5441
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Re: What will be the remainder when 13^36 is divided by 2196? [#permalink] New post   20 Aug 2017, 20:28
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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
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cbh
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Re: What will be the remainder when 13^36 is divided by 2196? [#permalink] New post   04 Sep 2017, 17:38
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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
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Re: What will be the remainder when 13^36 is divided by 2196? [#permalink] New post   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" :-D



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
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chetan2u
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Re: What will be the remainder when 13^36 is divided by 2196? [#permalink] New post   18 May 2018, 00:03
Expert Reply
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" :-D



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..
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Vidhi96
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Re: What will be the remainder when 13^36 is divided by 2196? [#permalink] New post   11 Jul 2020, 08:01
When ever you see a situation like this, try to bring numerator as close as denominator so,

(13)^3

=2197/2196 R will always be 1 irrespective the number of power

= (1)^12

= 1 answer
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Re: What will be the remainder when 13^36 is divided by 2196? [#permalink] New post   24 Jul 2023, 21:39
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Re: What will be the remainder when 13^36 is divided by 2196? [#permalink]
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