Re: What is the unit digit of 17^27?
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21 Feb 2024, 05:00
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).