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 EasyUnit digit of 3^100 is:ExplanationCycle length 4 for 3. 100 mod 4 = 0 → use the fourth value in the cycle → unit digit 1.
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Q2 EasyCycle length of unit digits of powers of 3 is:ExplanationPowers of 3 end in 3, 9, 7, 1 and then repeat → cycle length 4.
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Q3 EasyUnit digit of 9^77 is:ExplanationCycle of 9: 9, 1. Odd exponent → unit digit 9.
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Q4 EasyWhat unit digit appears at the end of 8 raised to the 44th power when cyclicity of powers is applied?ExplanationCycle of 8: 8, 4, 2, 6. 44 mod 4 = 0 → fourth position → unit digit 6.
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Q5 EasyWhat is the cyclicity length of the repeating unit-digit pattern for powers of the number 2?ExplanationPowers of 2 end in 2, 4, 8, 6 and repeat → cycle length 4.
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Q6 MediumUnit digit of 7^53 is:ExplanationCycle of 7: 7, 9, 3, 1. 53 mod 4 = 1 → first position → unit digit 7.
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Q7 MediumWhen a placement candidate evaluates the unit digit of 7 raised to the 101st power, which single digit should appear at the ones place?Explanation101 mod 4 = 1. In the cycle 7, 9, 3, 1, the first position gives unit digit 7.
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Q8 MediumA test asks for the remainder obtained when 3 raised to the 50th power is divided by 10; what is that remainder?ExplanationRemainder on division by 10 equals the unit digit. 50 mod 4 = 2 → second value in cycle 3, 9, 7, 1 → 9.
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Q9 MediumIf you need the unit digit of 13 raised to the 100th power, which digit should you select from the answer choices?ExplanationOnly the unit digit of the base matters: 3^100. 100 mod 4 = 0 → unit digit 1.
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Q10 MediumWhat unit digit results when 4 is raised to the 73rd power according to its two-digit cycle?ExplanationCycle of 4: 4, 6. Odd exponent → unit digit 4.
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Q11 MediumWhen 2 is raised to the 99th power, what digit appears in the units place of the result?ExplanationCycle 2, 4, 8, 6. 99 mod 4 = 3 → third value → unit digit 8.
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Q12 HardThe unit digit in 7^95 − 3^58 is:Explanation95 mod 4 = 3 → unit digit of 7^95 is 3. 58 mod 4 = 2 → unit digit of 3^58 is 9. Borrowing: 13 − 9 = 4.
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Q13 HardThe unit digit in (6374)^1793 × (625)^317 × (341)^491 is:ExplanationUnit digits are 4, 5, and 1. 4^1793 ends in 4 (odd power), 5^317 ends in 5, and 1^491 ends in 1. Product unit digit: 4 × 5 × 1 = 20 → 0.
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Q14 HardWhat remainder do you get when 7 raised to the 20th power is divided by 5, using cyclicity of remainders?Explanation7 mod 5 = 2. Powers of 2 mod 5 cycle as 2, 4, 3, 1. 20 mod 4 = 0 → remainder 1.
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DS1 Easy
What is the unit digit of 8 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 equal to 44.
ExplanationIf n is a multiple of 4 (including n = 44), 8^n ends in 6. Either statement alone is sufficient. -
DS2 Medium
What is the unit digit of 2 raised to the power n, where n is a positive integer?
Statement IThe remainder when n is divided by 4 is equal to 2.
Statement IIThe cycle length of the unit digits of powers of 2 is 4.
Explanationn mod 4 = 2 places 2^n at the second position in the cycle 2, 4, 8, 6 → unit digit 4. Knowing the cycle length alone does not fix n mod 4.
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DS3 Medium
What is the remainder when 3 raised to the power n is divided by 10?
Statement IThe value of n is equal to 50.
Statement IIThe value of n is an even positive integer.
Explanation3^50: 50 mod 4 = 2 → unit digit 9, so remainder is 9. Even n alone does not fix the remainder (e.g. n = 2 gives 9, n = 4 gives 1). -
DS4 Hard
What is the cycle length for the unit digits of powers of 7?
Statement IThe unit digit of 7 raised to the first power is 7.
Statement IIThe unit digit of 7 raised to the fifth power is also 7.
ExplanationIndividual powers ending in 7 do not prove the cycle length. You need the full repeating pattern (7, 9, 3, 1) to confirm length 4.