Quantitative Techniques · Micro-test
LCM and HCF
The HCF is the largest number that divides into a set of numbers; the LCM is the smallest number they all divide into. Most questions are about deciding which of the two the situation is asking for.
10 questions · 5 minutes · instant scoring
What this topic actually tests
Write each number as a product of primes. The HCF takes each prime that appears in every number, at its lowest power; the LCM takes every prime that appears in any number, at its highest power. For 12 = 2² × 3 and 18 = 2 × 3², the HCF is 2 × 3 = 6 and the LCM is 2² × 3² = 36. For two numbers, HCF × LCM = the product of the numbers, which lets you find a missing number or a missing LCM. Things that happen together again (bells, lights, runners on a track) call for the LCM of the intervals. Cutting or dividing into the largest equal pieces calls for the HCF. Remainder questions shift the numbers first: the largest number dividing 70 and 125 to leave remainders 5 and 8 is the HCF of 65 and 117. The smallest number leaving the same remainder r when divided by several numbers is their LCM plus r. For fractions, the HCF is the HCF of the numerators over the LCM of the denominators, and the LCM is the LCM of the numerators over the HCF of the denominators.
The common trap on this topic
The trap is reaching for the wrong one. 'When will they next happen together?' wants the LCM, a number larger than each interval; 'the largest size that fits exactly' wants the HCF, a number smaller than each length. If your answer is smaller than the numbers in a 'together again' question, or larger in a 'largest piece' question, you have used the wrong one. In remainder questions, subtract the remainders before finding the HCF, not after.
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What is the HCF of 36 and 84?
What is the LCM of 12, 18 and 30?
The product of two numbers is 2160 and their HCF is 12. What is their LCM?
Three bells ring at intervals of 6, 8 and 12 minutes. They ring together at 9:00 am. When will they next ring together?
What is the largest number that divides 70 and 125 leaving remainders 5 and 8 respectively?
What is the smallest number that leaves a remainder of 4 when divided by 12, by 15 and by 20?
The HCF of two numbers is 8 and their LCM is 48. One of the numbers is 16. What is the other?
Two numbers are in the ratio 3 : 4 and their LCM is 84. What is their sum?
A floor measures 4.8 m by 3.6 m. What is the least number of identical square tiles that will cover it exactly?
What is the HCF of 2/3, 8/9 and 16/81?
LCM and HCF: answers and explanations
What is the HCF of 36 and 84?
Answer: D. 12
36 = 2² × 3² and 84 = 2² × 3 × 7. The common primes at their lowest powers are 2² × 3 = 12.
What is the LCM of 12, 18 and 30?
Answer: D. 180
12 = 2² × 3, 18 = 2 × 3², 30 = 2 × 3 × 5. Every prime at its highest power: 2² × 3² × 5 = 180.
The product of two numbers is 2160 and their HCF is 12. What is their LCM?
Answer: B. 180
HCF × LCM = product, so LCM = 2160 ÷ 12 = 180.
Three bells ring at intervals of 6, 8 and 12 minutes. They ring together at 9:00 am. When will they next ring together?
Answer: A. 9:24 am
They coincide every LCM(6, 8, 12) = 24 minutes, so next at 9:24 am.
What is the largest number that divides 70 and 125 leaving remainders 5 and 8 respectively?
Answer: B. 13
The number divides 70 − 5 = 65 and 125 − 8 = 117 exactly. HCF(65, 117) = 13.
What is the smallest number that leaves a remainder of 4 when divided by 12, by 15 and by 20?
Answer: C. 64
LCM(12, 15, 20) = 60 divides exactly; add the remainder: 60 + 4 = 64.
The HCF of two numbers is 8 and their LCM is 48. One of the numbers is 16. What is the other?
Answer: B. 24
Other number = HCF × LCM ÷ first number = 8 × 48 ÷ 16 = 24.
Two numbers are in the ratio 3 : 4 and their LCM is 84. What is their sum?
Answer: C. 49
Write them as 3k and 4k. Their LCM is 12k = 84, so k = 7 and the numbers are 21 and 28. Their sum is 49.
A floor measures 4.8 m by 3.6 m. What is the least number of identical square tiles that will cover it exactly?
Answer: C. 12
The largest square tile has side HCF(480, 360) = 120 cm. The floor takes 480 ÷ 120 = 4 tiles one way and 360 ÷ 120 = 3 the other: 12 tiles.
What is the HCF of 2/3, 8/9 and 16/81?
Answer: B. 2/81
HCF of fractions = HCF of numerators ÷ LCM of denominators = HCF(2, 8, 16) ÷ LCM(3, 9, 81) = 2/81.
FAQ
What is the relation between LCM and HCF?
For any two numbers, HCF × LCM equals the product of the two numbers. It does not hold for three or more numbers.
When should I use LCM and when HCF?
Use the LCM when events repeat at different intervals and you want them to coincide. Use the HCF when you want the largest equal size that divides several quantities exactly.
How do I find the HCF of fractions?
Divide the HCF of the numerators by the LCM of the denominators.
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