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 EasyHow many factors does 48 have?Explanation48 = 2⁴ × 3. Number of factors = (4 + 1)(1 + 1) = 10.
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Q2 EasyHow many factors does 36 have?Explanation36 = 2² × 3². Number of factors = (2 + 1)(2 + 1) = 9.
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Q3 EasyHow many factors does 100 have?Explanation100 = 2² × 5². Number of factors = (2 + 1)(2 + 1) = 9.
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Q4 EasyHow many factors does 2^5 × 3^2 have?ExplanationNumber of factors = (5 + 1)(2 + 1) = 6 × 3 = 18.
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Q5 MediumHow many odd factors of 180 are there?Explanation180 = 2² × 3² × 5. Odd factors come from 3² × 5 only: (2 + 1)(1 + 1) = 6.
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Q6 MediumHow many factors of 64 are perfect squares?Explanation64 = 2⁶. Perfect-square factors use even exponents: 2⁰, 2², 2⁴, 2⁶ → 4 factors.
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Q7 MediumThe positive integer 72 can be written as 2³ × 3². How many distinct positive divisors does 72 have?Explanation72 = 2³ × 3². Number of factors = (3 + 1)(2 + 1) = 12.
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Q8 MediumWhich of the following numbers has exactly 9 factors?Explanation36 = 2² × 3² gives (2 + 1)(2 + 1) = 9 factors. The other options have 8, 10, and 12 factors respectively.
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Q9 MediumWhat is the total number of factors of 144?Explanation144 = 2⁴ × 3². Number of factors = (4 + 1)(2 + 1) = 15.
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Q10 MediumWhat is the smallest positive integer that has exactly 6 factors?Explanation6 factors can come from p⁵ or p² × q. The smallest such number is 2² × 3 = 12.
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Q11 MediumHow many composite factors does the number 24 have?Explanation24 has 8 factors in total. Excluding 1, 2, and 3, the remaining 5 factors are composite.
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Q12 HardHow many even factors does 120 have?Explanation120 = 2³ × 3 × 5 has 16 factors in total. It has 4 odd factors, so the number of even factors is 16 − 4 = 12.
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Q13 HardWhich number has exactly 5 factors?ExplanationA number with exactly 5 factors must be of the form p⁴. 16 = 2⁴ has 5 factors. 25 has only 3 factors.
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Q14 HardHow many perfect-square factors does 900 have?Explanation900 = 2² × 3² × 5². Perfect-square factors use even exponents only: (1 + 1)(1 + 1)(1 + 1) = 8.
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DS1 Easy
How many positive factors does the integer N have?
Statement IThe integer N is equal to 72.
Statement IIThe prime factorisation of N is 2³ × 3².
ExplanationBoth statements identify N = 72, which has 12 factors. Each statement alone is sufficient to answer the question. -
DS2 Medium
How many odd factors does the positive integer N have?
Statement IThe integer N is equal to 180.
Statement IIThe integer N is divisible by 5.
Explanation180 = 2² × 3² × 5 has 6 odd factors. Statement I alone is sufficient. Statement II alone does not fix N and allows many different factor counts.
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DS3 Medium
Does the positive integer N have exactly 6 factors?
Statement IThe integer N is equal to 12.
Statement IIThe integer N is the smallest positive number with exactly 6 factors.
Explanation12 has exactly 6 factors, so Statement I alone answers Yes. Statement II describes a property but does not by itself confirm the count without knowing N. -
DS4 Hard
How many perfect-square factors does the positive integer N have?
Statement IThe integer N is equal to 64.
Statement IIThe integer N is a power of 2.
Explanation64 = 2⁶ has 4 perfect-square factors. Statement I alone is sufficient. Statement II alone could refer to 2, 4, 8, 16, 32, or 64, each with a different count.