HCF and LCM
ICSE Class 6 · Mathematics · Chapter 8
What this chapter covers
- Factors, multiples and prime factorisation
- HCF: the Highest Common Factor
- LCM: the Lowest Common Multiple
- The property HCF × LCM = product of the two numbers
- Word problems on HCF and LCM
Key points
- Prime factorisation writes a number as a product of primes, for example 60 = 2 × 2 × 3 × 5.
- The HCF is the greatest number that divides all the given numbers exactly.
- For the HCF by prime factorisation, multiply only the primes common to all the numbers.
- For the HCF by division, keep dividing the last divisor by the remainder; the last divisor before remainder 0 is the HCF.
- The LCM is the smallest number, other than 0, that all the given numbers divide exactly.
- For the LCM by prime factorisation, take each prime the greatest number of times it appears in any one number.
- For the LCM by division, divide all the numbers together by primes until only 1s are left, then multiply the divisors.
- Co-prime numbers have HCF 1, and their LCM is their product.
- For two numbers, HCF × LCM = the product of the two numbers.
- Use the HCF for the biggest equal pieces or groups; use the LCM for the smallest common amount or the next time things happen together.
Questions answered
Write 90 as a product of prime factors.
90 = 2 × 3 × 3 × 5.
What is the HCF of 18 and 27? List the common factors to check.
The common factors are 1, 3 and 9, so the HCF is 9.
Find the HCF of 105 and 175 by the division method.
175 ÷ 105 gives remainder 70. 105 ÷ 70 gives remainder 35. 70 ÷ 35 gives remainder 0. So the HCF is 35.
Find the LCM of 8, 12 and 20.
8 = 2 × 2 × 2, 12 = 2 × 2 × 3 and 20 = 2 × 2 × 5. The LCM is 2 × 2 × 2 × 3 × 5 = 120.
Are 14 and 25 co-prime? What is their LCM?
Yes. Their only common factor is 1. So their LCM is their product, 350.
The product of two numbers is 1200 and their HCF is 10. What is their LCM?
LCM = 1200 ÷ 10 = 120.
Three bells ring every 6, 10 and 15 minutes. They ring together at 9:00 a.m. When will they next ring together?
The LCM of 6, 10 and 15 is 30, so they ring together again at 9:30 a.m.
A gardener has 56 marigold plants and 72 rose plants. He wants rows with the same number of plants, one kind in each row, as many in a row as possible. How many plants go in each row?
The HCF of 56 and 72 is 8, so each row has 8 plants. That gives 7 rows of marigolds and 9 rows of roses.