Practice Questions
HCF (Highest Common Factor)
The largest number that divides each of the given numbers exactly. Found via prime factorisation: take lowest powers of common primes.
LCM (Least Common Multiple)
The smallest number divisible by each given number. Take highest powers of all primes involved.
Key Relation
For two numbers a and b: HCF(a,b) × LCM(a,b) = a × b.
Methods
- Prime factorisation
- Division method (for HCF)
- Listing multiples (for small numbers)
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Q1 EasyWhat is the highest common factor of 24 and 36?Explanation24 = 2³ × 3 and 36 = 2² × 3². HCF takes the lowest powers of common primes: 2² × 3 = 12.
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Q2 EasyWhat is the least common multiple of 8 and 12?Explanation8 = 2³ and 12 = 2² × 3. LCM takes the highest powers of all primes involved: 2³ × 3 = 24.
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Q3 EasyWhat is the LCM of two co-prime numbers 7 and 11?ExplanationWhen two numbers are co-prime, their LCM equals their product: 7 × 11 = 77.
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Q4 MediumFind the greatest number that will divide 43, 91 and 183 so as to leave the same remainder in each case.ExplanationRequired number = H.C.F. of (91 − 43), (183 − 91) and (183 − 43) = H.C.F. of 48, 92 and 140 = 4.
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Q5 MediumThe H.C.F. of two numbers is 23 and the other two factors of their L.C.M. are 13 and 14. The larger of the two numbers is:ExplanationClearly, the numbers are (23 × 13) and (23 × 14). Larger number = (23 × 14) = 322.
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Q6 MediumThe greatest number of four digits which is divisible by 15, 25, 40 and 75 is:ExplanationL.C.M. of 15, 25, 40 and 75 is 600. On dividing 9999 by 600, the remainder is 399. Required number = (9999 − 399) = 9600.
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Q7 MediumThe largest 4 digit number exactly divisible by 88 is:ExplanationLargest 4-digit number = 9999. On dividing 9999 by 88, the remainder is 55. Required number = (9999 − 55) = 9944.
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Q8 MediumThe HCF of two numbers is 4 and their LCM is 60. If one of the numbers is 12, what is the other number?ExplanationProduct of two numbers = HCF × LCM. So 12 × other = 4 × 60 = 240, giving the other number as 20.
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Q9 MediumWhat is the HCF of 48, 72 and 96?Explanation48 = 2⁴ × 3, 72 = 2³ × 3², 96 = 2⁵ × 3. HCF = 2³ × 3 = 24.
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Q10 MediumWhat is the least number which when divided by 6, 8 and 12 leaves remainder 0 in each case?ExplanationThe required number is the LCM of 6, 8 and 12, which is 24.
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Q11 MediumThe product of two numbers is 216 and their HCF is 6. What is their LCM?ExplanationLCM = Product ÷ HCF = 216 ÷ 6 = 36.
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Q12 HardSix bells commence tolling together and toll at intervals of 2, 4, 6, 8 10 and 12 seconds respectively. In 30 minutes, how many times do they toll together ?ExplanationL.C.M. of 2, 4, 6, 8, 10, 12 is 120. So the bells toll together after every 120 seconds (2 minutes). In 30 minutes, they toll together (30 ÷ 2) + 1 = 16 times.
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Q13 HardLet N be the greatest number that will divide 1305, 4665 and 6905, leaving the same remainder in each case. Then sum of the digits in N is:ExplanationN = H.C.F. of (4665 − 1305), (6905 − 4665) and (6905 − 1305) = H.C.F. of 3360, 2240 and 5600 = 1120. Sum of digits in N = (1 + 1 + 2 + 0) = 4.
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Q14 HardTwo numbers are in the ratio 3 : 4 and their LCM is 84. What are the two numbers?ExplanationLet the numbers be 3k and 4k. LCM(3k, 4k) = 12k = 84, so k = 7. The numbers are 21 and 28.
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DS1 Easy
Two positive integers A and B are given. What is the HCF of A and B?
Statement IA equals 48 and B equals 72.
Statement IIThe highest common factor of A and B is 24.
ExplanationStatement I alone fixes A and B, so HCF(48, 72) = 24. Statement II states the answer directly but does not identify A and B on its own. -
DS2 Easy
What is the least common multiple of 9 and 12?
Statement IThe LCM of 9 and 12 is 36.
Statement IIThe numbers 9 and 12 are co-prime (they share no common factor other than 1).
ExplanationStatement I alone gives LCM = 36. Statement II is false because 9 and 12 share the factor 3, so it cannot be used.
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DS3 Medium
Two positive integers are in the ratio 2 : 3. What is their LCM?
Statement IThe two numbers are 14 and 21.
Statement IIThe HCF of the two numbers is 7.
ExplanationFrom ratio 2 : 3 with HCF 7, the numbers are 14 and 21, so LCM = 42. Neither statement alone fixes both numbers; together they do. -
DS4 Medium
Three positive integers are given. What is their HCF?
Statement IThe three numbers are 24, 36 and 60.
Statement IIEach of the three numbers is divisible by 12.
ExplanationStatement I alone gives HCF(24, 36, 60) = 12. Statement II alone only shows 12 is a common factor; the HCF could be 12 or a proper divisor of 12 without the actual numbers.