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 Medium
Unit digit of 7^53 is:
ExplanationCycle of 7: 7,9,3,1. 53 mod 4 = 1 → first in cycle: 7. -
Q2 Easy
Unit digit of 3^100 is:
ExplanationCycle length 4. 100 mod 4 = 0 → use 4th position: 1. -
Q3 Medium
Unit digit of 2^78 is:
Explanation2 cycle: 2,4,8,6. 78 mod 4 = 2 → unit digit 4. -
Q4 Easy
Unit digit of 9^77 is:
Explanation9 cycle: 9,1. Odd power → 9. -
Q5 Easy
Unit digit of 4^25 is:
Explanation4 cycle: 4,6. 25 odd → 4. -
Q6 Medium
Unit digit of (3^4 × 7^2) is:
Explanation3^4 unit 1, 7^2 unit 9. 1×9 = 9. -
Q7 Medium
Unit digit of 8^65 is:
Explanation8 cycle: 8,4,2,6. 65 mod 4 = 1 → 8.
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Q8 Easy
Unit digit of 12^34 is:
ExplanationOnly the last digit of the base matters: 2^34. Cycle 2,4,8,6; 34 mod 4 = 2 → unit digit 4. -
Q9 Easy
Unit digit of 6^99 is:
ExplanationAny positive power of 6 ends in 6. -
Q10 Hard
Unit digit of (7^3 + 3^7) is:
Explanation7^3 has unit digit 3. 3^7: 7 mod 4 = 3 → unit digit 7. 3 + 7 = 10 → unit digit 0. -
Q11 Medium
Unit digit of 13^45 is:
ExplanationSame as 3^45. 45 mod 4 = 1 → unit 3. -
Q12 Hard
Unit digit of 2^200 + 3^200 is:
Explanation2^200 has unit digit 6; 3^200 has unit digit 1. 6 + 1 = 7. -
Q13 Easy
Unit digit of 5^100 is:
ExplanationAny positive power of 5 ends in 5. -
Q14 Medium
Unit digit of 17^23 is:
ExplanationSame as 7^23. 23 mod 4 = 3 → cycle position 3 → unit 3.
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DS1 Medium
What is the unit digit of 7^n?
Statement In = 43.
Statement IIn is odd.
Explanation7^43: 43 mod 4 = 3 → unit digit 3. Statement II alone (n odd) gives unit digit 7 or 3 depending on n mod 4. -
DS2 Easy
What is the unit digit of 3^n?
Statement In is a multiple of 4.
Statement IIn > 10.
ExplanationIf n is a multiple of 4, 3^n has unit digit 1. Statement II alone does not fix the unit digit.
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DS3 Medium
What is the unit digit of a^b?
Statement Ia ends in 3.
Statement IIb = 25.
ExplanationUnit digit of 3^25: 25 mod 4 = 1 → 3. Neither statement alone gives both base and exponent. -
DS4 Easy
What is the unit digit of 9^n?
Statement In is odd.
Statement IIn = 77.
ExplanationEither statement alone suffices: odd n → unit digit 9; n = 77 (odd) → unit digit 9.