1. **State the problem:** We are given two numbers: $$A = 2^3 \times 7^2$$ $$B = 2^3 \times 5^3 \times 7$$ We need to find: (a) The highest common factor (HCF) of A and B. (b) The lowest common multiple (LCM) of A and B. 2. **Formulas and rules:** - The HCF of two numbers is found by taking the product of the lowest powers of all prime factors common to both numbers. - The LCM of two numbers is found by taking the product of the highest powers of all prime factors present in either number. 3. **Find the HCF:** - Prime factors of A: $2^3$, $7^2$ - Prime factors of B: $2^3$, $5^3$, $7^1$ - Common prime factors: $2$ and $7$ - Take the lowest powers: - For $2$: $\min(3,3) = 3$ - For $7$: $\min(2,1) = 1$ - So, $$\text{HCF} = 2^3 \times 7^1 = 8 \times 7 = 56$$ 4. **Find the LCM:** - Prime factors involved: $2$, $5$, $7$ - Take the highest powers: - For $2$: $\max(3,3) = 3$ - For $5$: $\max(0,3) = 3$ - For $7$: $\max(2,1) = 2$ - So, $$\text{LCM} = 2^3 \times 5^3 \times 7^2 = 8 \times 125 \times 49$$ - Calculate stepwise: - $8 \times 125 = 1000$ - $1000 \times 49 = 49000$ **Final answers:** - (a) HCF = 56 - (b) LCM = 49000