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What is the unit digit of 17^27?
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19 Feb 2024, 00:03
19 Feb 2024, 00:03
Expert Reply
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Question Stats:
66% (00:48) correct
33% (01:06) wrong based on 30 sessions
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What is the unit digit of \(17^{27}\)?
A. 1
B. 2
C. 3
D. 7
E. 9
A. 1
B. 2
C. 3
D. 7
E. 9
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Re: What is the unit digit of 17^27?
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21 Feb 2024, 05:00
21 Feb 2024, 05:00
Expert Reply
OE
When raising numbers to powers, there are always patterns for the units digit of the answer, and that pattern is based solely on the units digit of the starting base (the number that is not the exponent). In this problem, the base is 17 and the units digit of 17 is 7. So asking for the units digit of 17^27 is the same as asking for the units digit of 7^27.
But what is the units digit of 7^27? Find the pattern:
7^1 = 7
7^2 = 49 (the units digit is 9)
7^3 = 7 × 49 = ugh without a calculator
But it's still the case that you can find the units digit of any multiplication by multiplying only the units digits. In this case, multiply 9 × 7 to get 63. So the units digit of 7^3 is 3.
7^4 = 7 × some number that ends in 3. Use the units digits: 7 × 3 = 21, so the units digit of 7^4 is 1.
7^5 = 7 × some number that ends in 1, so the units digit of 7^5 is 7 × 1 = 7.
Stop here. The units digit of 7^5 matches the units digit of 7^1—both times the units digit is 7. So you've just found where the pattern repeats. It's a 4-step pattern:
7^1 = units digit of 7
7^2 = units digit of 9
7^3 = units digit of 3
7^4 = units digit of 1
That is, the pattern is 7, 9, 3, 1, repeat.
So how does 7^27 fit into this pattern? Use multiples to get closer: Seven to the fourth power has units digit of 1. So does seven to the eighth power and the twelfth power and all the way up to seven to the twenty-fourth power, which is the multiple of four that is closest to 27 without going over:
7^4 = units digit of 1
7^24 = units digit of 1
Follow the pattern from here:
7^24 = units digit of 1
7^25 = units digit of 7
7^26 = units digit of 9
7^27 = units digit of 3
The correct answer is (C).
When raising numbers to powers, there are always patterns for the units digit of the answer, and that pattern is based solely on the units digit of the starting base (the number that is not the exponent). In this problem, the base is 17 and the units digit of 17 is 7. So asking for the units digit of 17^27 is the same as asking for the units digit of 7^27.
But what is the units digit of 7^27? Find the pattern:
7^1 = 7
7^2 = 49 (the units digit is 9)
7^3 = 7 × 49 = ugh without a calculator
But it's still the case that you can find the units digit of any multiplication by multiplying only the units digits. In this case, multiply 9 × 7 to get 63. So the units digit of 7^3 is 3.
7^4 = 7 × some number that ends in 3. Use the units digits: 7 × 3 = 21, so the units digit of 7^4 is 1.
7^5 = 7 × some number that ends in 1, so the units digit of 7^5 is 7 × 1 = 7.
Stop here. The units digit of 7^5 matches the units digit of 7^1—both times the units digit is 7. So you've just found where the pattern repeats. It's a 4-step pattern:
7^1 = units digit of 7
7^2 = units digit of 9
7^3 = units digit of 3
7^4 = units digit of 1
That is, the pattern is 7, 9, 3, 1, repeat.
So how does 7^27 fit into this pattern? Use multiples to get closer: Seven to the fourth power has units digit of 1. So does seven to the eighth power and the twelfth power and all the way up to seven to the twenty-fourth power, which is the multiple of four that is closest to 27 without going over:
7^4 = units digit of 1
7^24 = units digit of 1
Follow the pattern from here:
7^24 = units digit of 1
7^25 = units digit of 7
7^26 = units digit of 9
7^27 = units digit of 3
The correct answer is (C).
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What is the unit digit of 17^27?
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03 Oct 2024, 06:37
03 Oct 2024, 06:37
1
What is the units digit of \(17^{27}\)?
When we need to find units digit of any power then units digit is same as the units digit of the base raised to the exponent.
=> Units digit of \(17^{27}\) = Units digit of \(7^{27}\)
Now to find the unit's digit of \(7^{27}\), we need to find the pattern / cycle of unit's digit of power of 7 and then generalizing it.
Unit's digit of \(7^1\) = 7
Unit's digit of \(7^2\) = 9
Unit's digit of \(7^3\) = 3
Unit's digit of \(7^4\) = 1
Unit's digit of \(7^5\) = 7
So, unit's digit of power of 7 repeats after every \(4^{th}\) number.
=> We need to divided 27 by 4 and check what is the remainder
=> 27 divided by 4 gives 3 remainder
=> \(7^{27}\) will have the same unit's digit as \(7^3\)
=> Unit's digits of \(7^{27}\) = 3
So, Answer will be C
Hope it helps!
Link to Theory for Last Two digits of exponents here.
Link to Theory for Units' digit of exponents here.
When we need to find units digit of any power then units digit is same as the units digit of the base raised to the exponent.
=> Units digit of \(17^{27}\) = Units digit of \(7^{27}\)
Now to find the unit's digit of \(7^{27}\), we need to find the pattern / cycle of unit's digit of power of 7 and then generalizing it.
Unit's digit of \(7^1\) = 7
Unit's digit of \(7^2\) = 9
Unit's digit of \(7^3\) = 3
Unit's digit of \(7^4\) = 1
Unit's digit of \(7^5\) = 7
So, unit's digit of power of 7 repeats after every \(4^{th}\) number.
=> We need to divided 27 by 4 and check what is the remainder
=> 27 divided by 4 gives 3 remainder
=> \(7^{27}\) will have the same unit's digit as \(7^3\)
=> Unit's digits of \(7^{27}\) = 3
So, Answer will be C
Hope it helps!
Link to Theory for Last Two digits of exponents here.
Link to Theory for Units' digit of exponents here.
