c) Consider the numbers 18 and 24 . (1) Find the product of these numbers. (2) Find the HCF and LCM of these numbers. (3) Find the product of the HCF and LCM. (4) What can you conclude about the product of two numbers \( a \) and \( b \) product of the HCF and LCM of \( a \) and \( b \) ? (1) Determine the HCF of two numbers if the product of the two numbe 2700 and the LCM is 180 .

Certainly! Let's tackle each part of the problem step by step. --- ### **Given Numbers:** 18 and 24 --- ### **1. Find the Product of These Numbers** **Product = 18 × 24** \[ 18 \times 24 = 432 \] **Answer:** The product of 18 and 24 is **432**. --- ### **2. Find the HCF and LCM of These Numbers** **a. Highest Common Factor (HCF) or Greatest Common Divisor (GCD):** **Method: Prime Factorization** 1. **Prime Factors of 18:** \[ 18 = 2 \times 3 \times 3 = 2 \times 3^2 \] 2. **Prime Factors of 24:** \[ 24 = 2 \times 2 \times 2 \times 3 = 2^3 \times 3 \] 3. **Common Prime Factors:** - **2:** Minimum power is \(2^1\) - **3:** Minimum power is \(3^1\) 4. **HCF:** \[ HCF = 2^1 \times 3^1 = 2 \times 3 = 6 \] **b. Least Common Multiple (LCM):** **Method: Prime Factorization** 1. **Combine Highest Powers of All Prime Factors:** - **2:** Maximum power is \(2^3\) - **3:** Maximum power is \(3^2\) 2. **LCM:** \[ LCM = 2^3 \times 3^2 = 8 \times 9 = 72 \] **Answer:** - **HCF (GCD) of 18 and 24 is **6**. - **LCM of 18 and 24 is **72**. --- ### **3. Find the Product of the HCF and LCM** \[ HCF \times LCM = 6 \times 72 = 432 \] **Answer:** The product of the HCF and LCM is **432**. --- ### **4. Conclusions About the Relationship Between Product, HCF, and LCM** From the above calculations: \[ \text{Product of the two numbers} = \text{HCF} \times \text{LCM} \] \[ 432 = 6 \times 72 \] **General Conclusion:** For any two positive integers \( a \) and \( b \): \[ a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b) \] **Answer:** *The product of two numbers is equal to the product of their HCF and LCM. In other words, for any two numbers \( a \) and \( b \), \[ a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b) \]* --- ### **5. Determine the HCF of Two Numbers Given Their Product and LCM** **Given:** - **Product of the two numbers (\( a \times b \)) = 2700** - **LCM (\( \text{LCM}(a, b) \)) = 180** **We know from the relationship:** \[ a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b) \] **Plugging in the known values:** \[ 2700 = \text{HCF}(a, b) \times 180 \] **Solve for HCF:** \[ \text{HCF}(a, b) = \frac{2700}{180} = 15 \] **Answer:** The HCF of the two numbers is **15**. ---