Practice Questions
Concept Explanation
A factor divides a number exactly. A multiple is obtained by multiplying the number by a positive integer.
Example: Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Multiples of 6 are 6, 12, 18, 24, …
Key Facts
- Every number has at least two factors: 1 and itself.
- 1 is neither prime nor composite.
- If a divides b and b divides c, then a divides c.
- Common multiples help in LCM problems; common factors help in HCF problems.
Placement Tips
List factors systematically in pairs (1×n, 2×…). For large numbers, use prime factorisation to count factors quickly.
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Q1 EasyThe sum of first 45 natural numbers is:ExplanationSum = n(n + 1)/2 = 45 × 46/2 = 1035.
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Q2 EasyHow many positive factors does the number 36 have?Explanation36 = 2² × 3². Number of factors = (2+1)(2+1) = 9.
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Q3 EasyWhich of the following is a factor of 84?Explanation84 ÷ 21 = 4 exactly, so 21 is a factor. The others leave a remainder.
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Q4 EasyWhat is the 8th multiple of 7?ExplanationThe 8th multiple of 7 is 7 × 8 = 56.
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Q5 MediumHow many 3-digit numbers are completely divisible by 6?ExplanationRequired numbers form an A.P.: 102, 108, …, 996 with a = 102, d = 6, l = 996. Number of terms n = 150.
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Q6 MediumHow many natural numbers are there between 23 and 100 which are exactly divisible by 6?ExplanationMultiples are 24, 30, 36, …, 96 — an A.P. with 13 terms.
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Q7 MediumIf n is a natural number, then (6n² + 6n) is always divisible by:Explanation(6n² + 6n) = 6n(n + 1). Since n(n + 1) is always even, the expression is divisible by both 6 and 12.
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Q8 MediumHow many multiples of 4 lie strictly between 11 and 49?ExplanationMultiples: 12, 16, 20, 24, 28, 32, 36, 40, 44, 48. Count = (48 − 12)/4 + 1 = 10.
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Q9 MediumWhich of the following numbers has exactly 3 factors?ExplanationOnly squares of primes have exactly 3 factors: 1, p, p². 9 = 3² has factors 1, 3, 9.
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Q10 MediumWhat is the sum of all factors of 12?ExplanationFactors of 12: 1, 2, 3, 4, 6, 12. Sum = 28.
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Q11 MediumHow many numbers less than 100 are multiples of both 3 and 5?ExplanationMultiples of lcm(3,5) = 15 below 100: 15, 30, 45, 60, 75, 90 → 6 numbers.
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Q12 HardHow many of the following numbers are divisible by 132? 264, 396, 462, 792, 968, 2178, 5184, 6336Explanation132 = 4 × 3 × 11. Numbers divisible by all three are 264, 396, 792 and 6336 — four in total.
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Q13 HardHow many of the following numbers are divisible by 3 but not by 9? 2133, 2343, 3474, 4131, 5286, 5340, 6336, 7347, 8115, 9276ExplanationNumbers divisible by 3 but not by 9 are 2343, 5286, 5340, 7347, 8115 and 9276 — six in total.
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Q14 HardHow many even factors does 60 have?Explanation60 = 2² × 3 × 5. Total factors = 12. Odd factors from 3 × 5 = 4. Even factors = 12 − 4 = 8.
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DS1 Easy
In a campus test, you need the factor count of N. How many positive factors does N have?
Statement IN = 2³ × 5.
Statement IIN is a perfect square.
ExplanationFrom the prime form, number of factors = (3+1)(1+1) = 8. Statement II alone does not identify N. -
DS2 Medium
Is the positive integer N divisible by 12?
Statement IN is a multiple of 6.
Statement IIN is a multiple of 4.
ExplanationN being a multiple of 6 does not guarantee divisibility by 4 (e.g. 6). N being a multiple of 4 does not guarantee divisibility by 3 (e.g. 4). Both together: lcm(6,4) = 12, so N is a multiple of 12.
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DS3 Medium
A revision sheet asks for the sum of all factors of N. What is the sum of factors of N?
Statement IN = 36.
Statement IIN is divisible by 9.
ExplanationN = 36 alone gives sum of factors = 91. Statement II with N = 9 gives sum 13 but N is not fixed to 36. -
DS4 Hard
Given clues about N in a reasoning set, is the positive integer N greater than 50?
Statement IN has 6 factors.
Statement IIN is a multiple of 13.
ExplanationSix factors could be 12, 18, 20, 28, 32, 44, etc. Multiples of 13 include 13, 26, 39, 52… Neither alone nor together fixes whether N > 50.