1. **Problem Statement:** (a) Find the Highest Common Factor (HCF) of 54 and 90. (b) Find the Lowest Common Multiple (LCM) of 54 and 90. 2. **Formulas and Rules:** - The HCF (or GCD) of two numbers is the greatest number that divides both without leaving a remainder. - The LCM of two numbers is the smallest number that is a multiple of both. - Important relation: $$\text{HCF}(a,b) \times \text{LCM}(a,b) = a \times b$$ 3. **Step-by-step solution for (a) HCF:** - Prime factorize 54: $$54 = 2 \times 3^3$$ - Prime factorize 90: $$90 = 2 \times 3^2 \times 5$$ - Common prime factors with lowest powers: $$2^1 \times 3^2 = 2 \times 9 = 18$$ - So, $$\text{HCF}(54,90) = 18$$ 4. **Step-by-step solution for (b) LCM:** - Use the relation: $$\text{LCM}(54,90) = \frac{54 \times 90}{\text{HCF}(54,90)}$$ - Substitute values: $$\frac{54 \times 90}{18} = \frac{4860}{18} = 270$$ - So, $$\text{LCM}(54,90) = 270$$ **Final answers:** - HCF of 54 and 90 is **18**. - LCM of 54 and 90 is **270**.