They don't give you a calculator on this test and you have to average about 2 minutes on quant problems. So...start calculating frantically in order to find 17 raised to the 27th power?
No, please don't! The test writers are not trying to see whether you can memorize something to the power of 27. Whenever you see a calculation that would clearly be impossible to do without a calculator in just a couple of minutes, some kind of pattern is involved. Your task is to find that pattern.
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 1727 is the same as asking for the units digit of 727.
But what is the units digit of 727? Find the pattern:
71 = 7
72 = 49 (the units digit is 9)
73 = 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 73 is 3.
74 = 7 × some number that ends in 3. Use the units digits: 7 × 3 = 21, so the units digit of 74 is 1.
75 = 7 × some number that ends in 1, so the units digit of 75 is 7 × 1 = 7.
Stop here. The units digit of 75 matches the units digit of 71—both times the units digit is 7. So you've just found where the pattern repeats. It's a 4-step pattern:
71 = units digit of 7
72 = units digit of 9
73 = units digit of 3
74 = units digit of 1
That is, the pattern is 7, 9, 3, 1, repeat.
So how does 727 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:
74 = units digit of 1
724 = units digit of 1
Follow the pattern from here:
724 = units digit of 1
725 = units digit of 7
726 = units digit of 9
727 = units digit of 3
The correct answer is (C).
Tip: All units digits have repeating patterns of either 1 number, 2 numbers, or 4 numbers. As you saw, 7 had a 4-number pattern (7, 9, 3, 1, repeat). The units digit 6 has a 1-number pattern (6, 6, 6—it's always 6!). The units digit 9 has a 2-number pattern (9, 1, repeat). You can memorize these if you want or you can figure them out by calculating.
