Practice Questions
Unit Digit of Powers
The unit digit of a^b depends on the cyclicity of the base. Memorise cycles for 2, 3, 7, 8 (length 4) and 4, 9 (length 2).
Cyclicity Table
| Base | Unit digits cycle | Cycle length |
|---|---|---|
| 2 | 2, 4, 8, 6 | 4 |
| 3 | 3, 9, 7, 1 | 4 |
| 7 | 7, 9, 3, 1 | 4 |
| 9 | 9, 1 | 2 |
Find unit digit of a^b: compute b mod (cycle length), use that power in the cycle.
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Q1 EasyWhat is the unit digit when 3 is raised to the power 100?ExplanationThe unit-digit cycle of 3 is 3, 9, 7, 1 (length 4). Since 100 mod 4 = 0, use the 4th position: unit digit 1.
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Q2 EasyWhat is the unit digit when 9 is raised to the power 77?ExplanationThe unit-digit cycle of 9 is 9, 1 (length 2). Since 77 is odd, the unit digit is 9.
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Q3 EasyThe unit digit of 4 raised to the power 25 is which of the following?ExplanationThe unit-digit cycle of 4 is 4, 6. Since 25 is odd, the unit digit is 4.
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Q4 EasyWhat is the unit digit of 12 raised to the power 34?ExplanationOnly the last digit of the base matters: 2^34. Cycle 2, 4, 8, 6; 34 mod 4 = 2, so the unit digit is 4.
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Q5 MediumWhat is the unit digit in 7^53 ?ExplanationUnit digit in 7^53 = Unit digit in [(7^4)^13 × 7]. Unit digit in (7^4)^13 = 1. Therefore unit digit in 7^53 = (1 × 7) = 7.
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Q6 MediumWhat is the unit digit in 7^105 ?ExplanationUnit digit in 7^105 = Unit digit in [(7^4)^26 × 7]. Unit digit in (7^4)^26 = 1. Therefore unit digit in 7^105 = (1 × 7) = 7.
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Q7 MediumWhat is the unit digit in 7^95 ?ExplanationUnit digit in 7^95 = Unit digit in [(7^4)^23 × 7^3]. Unit digit in (7^4)^23 = 1 and unit digit in 7^3 = 3. Therefore the unit digit is 3.
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Q8 MediumWhat is the unit digit in 3^58 ?ExplanationUnit digit in 3^58 = Unit digit in [(3^4)^14 × 3^2]. Unit digit in (3^4)^14 = 1 and unit digit in 3^2 = 9. Therefore the unit digit is 9.
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Q9 MediumWhat is the unit digit of 2 raised to the power 78?ExplanationThe unit-digit cycle of 2 is 2, 4, 8, 6. Since 78 mod 4 = 2, the unit digit is 4.
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Q10 MediumWhen 8 is raised to the power 65, what is the unit digit of the result?ExplanationThe unit-digit cycle of 8 is 8, 4, 2, 6. Since 65 mod 4 = 1, the unit digit is 8.
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Q11 MediumWhat is the unit digit of 13 raised to the power 45?ExplanationOnly the last digit matters: same as 3^45. Since 45 mod 4 = 1, the unit digit is 3.
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Q12 MediumWhat is the unit digit of 17 raised to the power 23?ExplanationOnly the last digit matters: same as 7^23. Since 23 mod 4 = 3, the cycle position gives unit digit 3.
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Q13 HardWhat is the unit digit in (7^95 - 3^58)?ExplanationUnit digit in 7^95 = 3 and unit digit in 3^58 = 9. Unit digit in (7^95 − 3^58) = unit digit in (343 − 9) = unit digit in 334 = 4.
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Q14 HardWhat is the unit digit in {(6374)^1793 x (625)^317 x (341^491)}?ExplanationUnit digit in (6374)^1793 = 4, in (625)^317 = 5, and in (341)^491 = 1. Required digit = unit digit in (4 × 5 × 1) = 0.
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DS1 Easy
What is the unit digit of 3 raised to the power n, where n is a positive integer?
Statement IThe exponent n is a multiple of 4.
Statement IIThe exponent n is greater than 10.
ExplanationIf n is a multiple of 4, 3^n has unit digit 1. Statement II alone only restricts n > 10 and does not fix the unit digit. -
DS2 Easy
What is the unit digit of 9 raised to the power n, where n is a positive integer?
Statement IThe exponent n is an odd positive integer.
Statement IIThe exponent n equals 77.
ExplanationEither statement alone suffices: odd n gives unit digit 9, and n = 77 (odd) also gives unit digit 9.
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DS3 Medium
A positive integer n is given. What is the unit digit of 7 raised to the power n?
Statement IThe exponent n equals 43.
Statement IIThe exponent n is an odd positive integer.
Explanation7^43: 43 mod 4 = 3, so the unit digit is 3. Statement II alone (n odd) could give unit digit 7 or 3 depending on n mod 4. -
DS4 Medium
A positive integer a and a positive exponent b are given. What is the unit digit of a raised to the power b?
Statement IThe number a ends in the digit 3.
Statement IIThe exponent b equals 25.
ExplanationUnit digit of 3^25: 25 mod 4 = 1, so the unit digit is 3. Neither statement alone gives both the base unit digit and the exponent.