Practice Questions
Cyclicity Concept
Unit digits of powers repeat in cycles. Once the cycle length is known, reduce the exponent modulo cycle length.
Remainder Cyclicity
Remainders when dividing successive powers also repeat — useful for large exponent remainder problems.
Key Cycles
2,3,7,8 have cycle length 4. 4 and 9 have cycle length 2. 0,1,5,6 have fixed unit digits.
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Q1 Easy
Cycle length of unit digits of powers of 3 is:
Explanation3, 9, 7, 1 repeat → cycle length 4. -
Q2 Medium
Unit digit of 7^101 is:
Explanation101 mod 4 = 1 → unit digit 7. -
Q3 Medium
Remainder when 3^50 is divided by 10:
ExplanationUnit digit of 3^50: 50 mod 4 = 2 → 9. -
Q4 Easy
Unit digit of 8^44 is:
Explanation8 cycle: 8,4,2,6. 44 mod 4 = 0 → 6. -
Q5 Easy
Cycle length for unit digit of 9^n is:
Explanation9, 1 alternate → cycle 2. -
Q6 Medium
Unit digit of 2^99 is:
ExplanationCycle 2,4,8,6. 99 mod 4 = 3 → third value is 8. -
Q7 Medium
Remainder when 4^73 is divided by 10:
Explanation4 cycle 4,6. 73 odd → 4.
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Q8 Easy
Unit digit of 1^1000 is:
Explanation1 to any power is 1. -
Q9 Easy
Unit digit of 6^777 is:
Explanation6 to any positive power ends in 6. -
Q10 Hard
Remainder when 7^20 is divided by 5:
Explanation7 mod 5 = 2. 2^20 mod 5: cycle 2,4,3,1 length 4. 20 mod 4=0 → 1. -
Q11 Medium
Unit digit of 3^15 is:
Explanation15 mod 4 = 3 → 7. -
Q12 Easy
Cycle length of 2^n unit digits:
Explanation2,4,8,6 → length 4. -
Q13 Medium
Unit digit of 13^100 is:
ExplanationSame as 3^100. 100 mod 4 = 0 → unit 1. -
Q14 Hard
Remainder when 2^30 is divided by 7:
Explanation2^3 ≡ 1 (mod 7), so 2^30 = (2^3)^10 ≡ 1^10 = 1.
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DS1 Medium
What is the unit digit of 2^n?
Statement IThe cycle length of 2 is 4.
Statement IIn mod 4 = 2.
Explanationn mod 4 = 2 places 2^n at the second position in cycle 2,4,8,6 → unit digit 4. Statement I alone does not fix n mod 4. -
DS2 Medium
What is the remainder when 3^n is divided by 10?
Statement In = 50.
Statement IIn is even.
Explanation3^50 mod 10: 50 mod 4 = 2 → unit digit 9, remainder 9. Statement II alone insufficient.
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DS3 Hard
What is the cycle length for unit digits of 7?
Statement I7^1 ends in 7.
Statement II7^5 ends in 7.
ExplanationKnowing individual powers end in 7 does not prove cycle length without full cycle check. -
DS4 Easy
What is the unit digit of 8^n?
Statement In is a multiple of 4.
Statement IIn = 44.
Explanationn = 44 (multiple of 4) → unit digit 6. Either statement alone suffices.