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# What is the units digit of 13^35?

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Math Expert
Joined: 02 Sep 2009
Posts: 58427
What is the units digit of 13^35?  [#permalink]

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03 Feb 2019, 22:17
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Difficulty:

25% (medium)

Question Stats:

67% (00:39) correct 33% (00:54) wrong based on 73 sessions

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What is the units digit of 13^35?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

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Re: What is the units digit of 13^35?  [#permalink]

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03 Feb 2019, 22:33
1
Bunuel wrote:
What is the units digit of 13^35?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

We need to focus only on units digit of 13 and that is 3.

3^1 = 3 ; 3^2 = 9 ; 3^3 = 7 ; 3^4 = 1 ; 3^5 = 3

Cyclicity = 3, 9, 7, 1 (4 digits)

Divide 35 by 4 = 35/4 = 8 remainder 3

Means 4 complete cycles of 3, 9, 7, 1

Remainder 3 means unit digit is 7.

(D)

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Re: What is the units digit of 13^35?  [#permalink]

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03 Feb 2019, 22:38
1
unit digit of powers of three
$$3^1 = 3 \, \, \: | \, \, \: 3^5 = 3$$
$$3^2 = 9 \, \, \; | \, \, \: 3^6 = 9$$
$$3^3 = 7 \, \, \; | \, \, \: 3^7 = 7$$
$$3^4 = 1 \, \, \; | \, \, \: 3^8 = 1$$

Cyclicity of 3 is 4
$$3^{35}$$
35 when divided by four leaves remainder 3
Hence units digits is 7
IMO D
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Re: What is the units digit of 13^35?  [#permalink]

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04 Feb 2019, 00:07
1
Bunuel wrote:
What is the units digit of 13^35?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

Cyclicity: The concept when the unit digit repeats after a cycle of some powers
e.g. The cyclicity of unit digits for the number with unit digit 2, 3, 7 and 8 is 4
and The cyclicity of unit digits for the number with unit digit 0, 1, 5 and 6 is 1
and The cyclicity of unit digits for the number with unit digit 4 and 9 is 2

The unit digit of $$13^{35}$$ = Unit digit of $$3^{35}$$ for calculation of unit digit of a number, only it's unit digit is important and any other digit is useless

i.e. Unit digit of 3^35 = unit digit of 3^3 becasue thewhen 35 is divided by cyclicity number 4 then remainder is 3

unit digit of 3^3 = 7

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Re: What is the units digit of 13^35?  [#permalink]

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04 Feb 2019, 02:08
Bunuel wrote:
What is the units digit of 13^35?

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

using cyclicity rule we can solve as only unit digits value is to be determined
the units digit of 13^35 = 3^ 35 = 7
IMO D
Re: What is the units digit of 13^35?   [#permalink] 04 Feb 2019, 02:08
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