Practice Questions
Counting Factors
If N = p^a × q^b × r^c … then the number of positive factors of N is (a+1)(b+1)(c+1)…
Special Cases
- Perfect squares have an odd number of factors.
- Prime numbers have exactly 2 factors.
- Number of odd factors: exclude powers of 2 from factorisation.
Example
72 = 2³ × 3² → (3+1)(2+1) = 12 factors.
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Q1 Easy
How many factors does 48 have?
Explanation48 = 2⁴ × 3. Factors = (4+1)(1+1) = 10. -
Q2 Easy
How many factors does 100 have?
Explanation100 = 2² × 5². Factors = 3×3 = 9. -
Q3 Medium
Number of odd factors of 180 is:
Explanation180 = 2² × 3² × 5. Odd factors from 3²×5: (2+1)(1+1) = 6. -
Q4 Medium
How many factors of 64 are perfect squares?
Explanation64 = 2⁶. Square factors 2⁰,2²,2⁴,2⁶ → 4. -
Q5 Medium
The smallest number with exactly 6 factors is:
Explanation6 = 6 or 2×3. Smallest: 2²×3=12 or 2×5=32? 12 has 6 factors. -
Q6 Easy
How many factors does 2^5 × 3^2 have?
Explanation(5+1)(2+1) = 18. -
Q7 Hard
Which number has exactly 5 factors?
Explanation5 factors → form p^4. Smallest 2^4=16. 25 has 3 factors.
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Q8 Medium
Total factors of 144:
Explanation144 = 2⁴ × 3². (4+1)(2+1) = 15. -
Q9 Hard
How many even factors does 120 have?
Explanation120 = 2³×3×5, total 16 factors. Odd factors 4. Even = 16−4 = 12. -
Q10 Easy
Number of factors of 1 is:
Explanation1 has exactly one factor: itself. -
Q11 Hard
If N has 12 factors and N = 2^a × 3^b, a possible pair (a,b) is:
Explanation(a+1)(b+1)=12. (4)(3)=12 → a=3,b=2. -
Q12 Hard
How many factors of 900 are perfect squares?
Explanation900 = 2² × 3² × 5². Square factors use even exponents: (1+1)(1+1)(1+1) = 8. -
Q13 Easy
Factors of 30 include how many prime numbers?
ExplanationPrime factors 2, 3, 5 → 3 (as factors of 30). -
Q14 Medium
How many composite factors does 24 have?
Explanation24 has 8 factors total. Non-composite factors are 1, 2, 3. The remaining 5 factors are composite.
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DS1 Easy
How many factors does N have?
Statement IN = 72.
Statement IIN = 2³ × 3².
ExplanationEither form identifies N = 72 with 12 factors. Both statements alone are sufficient. -
DS2 Medium
How many odd factors does N have?
Statement IN = 180.
Statement IIN is divisible by 5.
Explanation180 = 2²×3²×5 → 6 odd factors. Statement II alone (e.g. 5, 10, 15…) does not fix N.
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DS3 Medium
Does N have exactly 6 factors?
Statement IN = 12.
Statement IIN is the smallest such number.
Explanation12 has exactly 6 factors. Statement II adds ordering but I alone answers yes. -
DS4 Hard
How many perfect-square factors does N have?
Statement IN = 64.
Statement IIN is a power of 2.
Explanation64 = 2⁶ → 4 square factors. Statement II alone could be 2, 4, 8, 16, 32, 64… with varying counts.