4. Find the HCF and LCM for th a) 40 and 56

To find the **Highest Common Factor (HCF)** and the **Least Common Multiple (LCM)** of the numbers **40** and **56**, we can use the **Prime Factorization Method**. Here's a step-by-step guide: --- ### **1. Prime Factorization** **a) Prime Factors of 40:** 1. **40 ÷ 2 = 20** 2. **20 ÷ 2 = 10** 3. **10 ÷ 2 = 5** 4. **5 ÷ 5 = 1** So, the prime factors of 40 are: \[ 40 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5 \] **b) Prime Factors of 56:** 1. **56 ÷ 2 = 28** 2. **28 ÷ 2 = 14** 3. **14 ÷ 2 = 7** 4. **7 ÷ 7 = 1** So, the prime factors of 56 are: \[ 56 = 2 \times 2 \times 2 \times 7 = 2^3 \times 7 \] --- ### **2. Finding the HCF (Highest Common Factor)** The **HCF** is the product of the **lowest powers** of all common prime factors. - **Common Prime Factors of 40 and 56:** \( 2^3 \) Therefore: \[ \text{HCF} = 2^3 = 8 \] --- ### **3. Finding the LCM (Least Common Multiple)** The **LCM** is the product of the **highest powers** of all prime factors present in either number. - **Prime Factors to Consider:** \( 2^3 \), \( 5 \), and \( 7 \) Therefore: \[ \text{LCM} = 2^3 \times 5 \times 7 = 8 \times 5 \times 7 = 280 \] *Alternatively, you can use the formula:* \[ \text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)} \] \[ \text{LCM}(40, 56) = \frac{40 \times 56}{8} = \frac{2240}{8} = 280 \] --- ### **Final Answer:** - **HCF of 40 and 56 is 8** - **LCM of 40 and 56 is 280**