krit.club logo

Number System - Highest Common Factor (HCF) and Lowest Common Multiple (LCM)

Grade 6IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Factors: Numbers that divide exactly into another number without leaving a remainder.

•

Multiples: Numbers found in a particular number's times table (e.g., multiples of 5 are 5, 10, 15...).

•

Prime Numbers: Numbers with exactly two factors: 1 and themselves.

•

Prime Factorization: Expressing a number as a product of its prime factors (e.g., 12=22×312 = 2^2 \times 3).

•

Highest Common Factor (HCF): The largest number that is a factor of two or more numbers.

•

Lowest Common Multiple (LCM): The smallest number that is a multiple of two or more numbers.

•

Venn Diagram Method: A visual way to find HCF and LCM by placing common prime factors in the intersection.

📐Formulae

Product of two numbers=HCF(a,b)×LCM(a,b)\text{Product of two numbers} = \text{HCF}(a, b) \times \text{LCM}(a, b)

HCF=Product of the lowest powers of common prime factors\text{HCF} = \text{Product of the lowest powers of common prime factors}

LCM=Product of the highest powers of all prime factors present\text{LCM} = \text{Product of the highest powers of all prime factors present}

💡Examples

Problem 1:

Find the HCF and LCM of 24 and 36 using prime factorization.

Solution:

  1. Prime factors of 24=2×2×2×3=23×324 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3.
  2. Prime factors of 36=2×2×3×3=22×3236 = 2 \times 2 \times 3 \times 3 = 2^2 \times 3^2.
  3. HCF = 22×3=4×3=122^2 \times 3 = 4 \times 3 = 12.
  4. LCM = 23×32=8×9=722^3 \times 3^2 = 8 \times 9 = 72.

Explanation:

To find the HCF, we take the lowest power of the common prime factors (2 and 3). To find the LCM, we take the highest power of every prime factor that appears in the factorizations.

Problem 2:

Two bells ring at intervals of 15 minutes and 20 minutes respectively. If they ring together at 10:00 AM, at what time will they next ring together?

Solution:

  1. Find LCM of 15 and 20.
  2. Multiples of 15: 15, 30, 45, 60, 75...
  3. Multiples of 20: 20, 40, 60, 80...
  4. LCM = 60.
  5. 60 minutes = 1 hour.
  6. Time: 10:00 AM + 1 hour = 11:00 AM.

Explanation:

This is a real-world application of LCM. Since we need to find the next 'common' event in the future, we look for the Lowest Common Multiple of the two time intervals.

Problem 3:

Given that the HCF of two numbers is 6 and their LCM is 72, if one of the numbers is 18, find the other number.

Solution:

  1. Use the formula: n1×n2=HCF×LCMn_1 \times n_2 = HCF \times LCM.
  2. 18×n2=6×7218 \times n_2 = 6 \times 72.
  3. 18×n2=43218 \times n_2 = 432.
  4. n2=432/18=24n_2 = 432 / 18 = 24.

Explanation:

We use the relationship between HCF, LCM, and the product of the two numbers to solve for the unknown value.