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What is the units digit of 17^83 × 13^82 × 11^87?

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Bunuel
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What is the units digit of 17^83 × 13^82 × 11^87? [#permalink] New post   22 Apr 2016, 04:29
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What is the units digit of 17^83 × 13^82 × 11^87?

A. 4
B. 5
C. 6
D. 7
E. 8
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D
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Re: What is the units digit of 17^83 × 13^82 × 11^87? [#permalink] New post   22 Apr 2016, 11:51
Cyclicity of 7 is 4 --> 7, 9, 3, 1

Cyclicity of 3 is 4 --> 3, 9, 7, 1

11 raised to any positive power has a units digit 1

17^83 × 13^82 × 11^87 = Units digit of 3 * 9 * 1 = 7

Answer: D
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Re: What is the units digit of 17^83 × 13^82 × 11^87? [#permalink] New post   22 Apr 2016, 15:43
To find : the units digit of 17^83 × 13^82 × 11^87

Let's reduce the clutter and simplify the product
(7^83) (3^82) (1^87)

7 has a cyclicity of 4 : the last digit of any positive power of 7 repeats itself after every 4th power

so 7^5 has the same last digit as 7^1, 7^9, 7^13
Thus , 7^83 has the same last digit as 7^3, 7^7, 7^11 i.e. 3


3 has a cyclicity of 4 : exactly the same routine as above
Thus , 3^82 has the same last digit as 3^2, 3^6, 3^10 i.e. 9

Any power of 1 will result in 1 as the last digit

So, product of our last digits = 3 x 9 x 1 = 27....last digit is 7

Correct Option : D
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Re: What is the units digit of 17^83 × 13^82 × 11^87? [#permalink] New post   19 Apr 2020, 09:28
What is the units digit of 17^83 × 13^82 × 11^87?

Cyclicity of 7 is 4 --> 7, 9, 3, 1
units digit of 17^83 = units digit of 17^3

Cyclicity of 3 is 4 --> 3, 9, 7, 1
units digit of 13^82 = units digit of 13^2

11 raised to any positive power has a units digit 1

the units digit of 17^83 × 13^82 × 11^87 = Units digit of 3 * 9 * 1 = 7

correct answer: D
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Re: What is the units digit of 17^83 × 13^82 × 11^87? [#permalink] New post   20 Apr 2020, 08:06
Bunuel wrote:
What is the units digit of 17^83 × 13^82 × 11^87?

A. 4
B. 5
C. 6
D. 7
E. 8


To find units digit we take the unit digit of the number required and cycle it out till it repeats itself.

7 = 7,49,343,2401 = Cycle of 4. Thus \(17^{83}\) will have units digit = \(4*20+3 = 3\)
3 = 3,9,27,81 = Cycle of 4. Thus \(13^{82}\) will have units digit = \(4*20+2 = 9\)
11 will always end in 1. Thus = 1

Final answer = \(3*9*1 = 27\)
Answer D
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Re: What is the units digit of 17^83 × 13^82 × 11^87? [#permalink]
20 Apr 2020, 08:06
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