🔢 Prime Time: Mastering HCF and LCM with the Power of Primes

The concepts of **Factors** and **Multiples** form the bedrock of the Number unit. While simple, they are crucial for tackling complex fractions, ratio problems, and real-world scenarios involving time or distance. Knowing the quickest method to find the **Highest Common Factor (HCF)** and the **Lowest Common Multiple (LCM)** is key to saving time in your IGCSE exam.

The Power of Prime Factorization

For medium or large numbers, the most reliable and efficient method is using **Prime Factorization**. Every number can be written as a product of prime numbers (e.g.,

12 = 2^2 \times 3

). This gives you a clear blueprint for HCF and LCM:

**HCF:** Find the product of the **lowest power** of all common prime factors.
**LCM:** Find the product of the **highest power** of all prime factors involved.

Worked Example: Finding HCF and LCM

Problem: Find the HCF and LCM of

120

and

150

**Step 1 (Prime Factors):**
$120 = 2^3 \times 3^1 \times 5^1$

$150 = 2^1 \times 3^1 \times 5^2$
**Step 2 (Calculate HCF):** Take the lowest power of the common factors ( $2^1, 3^1, 5^1$ ):
$\text{HCF} = 2^1 \times 3^1 \times 5^1 = \mathbf{30}$
**Step 3 (Calculate LCM):** Take the highest power of all factors ( $2^3, 3^1, 5^2$ ):
$\text{LCM} = 2^3 \times 3^1 \times 5^2 = 8 \times 3 \times 25 = \mathbf{600}$

Mastering this technique will ensure accuracy, especially when dealing with those tricky exam questions involving multiple factors. Ready to solve more problems? Join our live practice session this week!