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What is the units digit of 13^27?

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What is the units digit of 13^27?  [#permalink]

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New post 14 Jun 2015, 03:58
2
00:00
A
B
C
D
E

Difficulty:

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Question Stats:

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What is the units digit of \(13^{27}\)?

A. 1
B. 2
C. 3
D. 7
E. 9

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Re: What is the units digit of 13^27?  [#permalink]

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New post 14 Jun 2015, 07:20
3 has a cyclicity of 4 (3,9,7,1). So power 7 means 4+3 . That is 7
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What is the units digit of 13^27?  [#permalink]

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New post 06 Jun 2016, 12:20
Power of 3 follows repetition pattern of = 3, 9, 7, 1, 3, 9, 7, 1 ....
Repetition after 4 powers = (3, 9, 7, 1)
13^27 = 27th Power = 24+3
24/4 = 6 (6 repetitions) + 3 = "7" ... Answer D...

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Re: What is the units digit of 13^27?  [#permalink]

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New post 23 May 2017, 05:51
Although I marked wrong answer due to silly mistake, but below is the solution.

3*3 = 9
3*3*3 27
3*3*3*3 = 81
3*3*3*3*3 = 243
3*3*3*3*3*3 = 729


So cycle is (9,7,1,3,9........)

Cycle 1 : took \(3^5\) to end. In simple words \(3^6\) started cycle 2 [ Refer above]

We need to find \(13^{27}\).

\(\frac{27}{5}\) = 25 + 2 => 5 cycles complete + 2 steps more.

=> 27.

Hence, 7 is unit place digit.

Ans: D
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Re: What is the units digit of 13^27?  [#permalink]

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New post 23 May 2017, 08:31
reto wrote:
What is the units digit of \(13^{27}\)?

A. 1
B. 2
C. 3
D. 7
E. 9

Cyclicity of digit 3 is 4

Now, we have \(13^{27}\)

= \(13^{4*6}*13^3\)

= \(1*7\)

Thus, answer must be (D) 7

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Re: What is the units digit of 13^27?  [#permalink]

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New post 23 May 2017, 08:38
reto wrote:
What is the units digit of \(13^{27}\)?

A. 1
B. 2
C. 3
D. 7
E. 9


\(13^{27}=(13^2)^{13} \times 13 \\
=(...9)^{13} \times 13 = (...9)^{12} \times ((...9) \times 13) \\
= ((...9)^2)^6 \times (...7) =(...1)^6 \times (...7) \\
= (...1) \times (...7) = (...7)\)

The answer is D
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What is the units digit of 13^27?  [#permalink]

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New post 07 Jul 2017, 06:46
reto wrote:
What is the units digit of \(13^{27}\)?

A. 1
B. 2
C. 3
D. 7
E. 9


Theory:
The cyclicity of 2,3,7,8 is 4.
When Power is divided by cyclicity; Remainder is zero, then Last digit^cyclicity
When power is divided by cyclicity; Remainder is Non zero, then Last digit ^Remainder


27=4x6+3
remainder=3
Thus, 3^3=27
Hence, Unit digit =7

Ans is D
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What is the units digit of 13^27?   [#permalink] 07 Jul 2017, 06:46
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