1. **Problem statement:** We are given two numbers expressed as products of prime factors: $$T = 2^5 \times 3^8 \times 5$$ $$X = 2^4 \times 3^2 \times 5 \times 7$$ We need to find: a) The lowest common multiple (LCM) of $T$ and $X$. b) The highest common factor (HCF) of $T$ and $X$. 2. **Formulas and rules:** - The LCM of two numbers is found by taking the highest power of each prime factor present in either number. - The HCF of two numbers is found by taking the lowest power of each prime factor common to both numbers. 3. **Find the LCM:** - For prime factor 2: max power is $\max(5,4) = 5$ - For prime factor 3: max power is $\max(8,2) = 8$ - For prime factor 5: max power is $\max(1,1) = 1$ - For prime factor 7: max power is $\max(0,1) = 1$ So, $$\text{LCM} = 2^5 \times 3^8 \times 5^1 \times 7^1$$ 4. **Find the HCF:** - For prime factor 2: min power is $\min(5,4) = 4$ - For prime factor 3: min power is $\min(8,2) = 2$ - For prime factor 5: min power is $\min(1,1) = 1$ - For prime factor 7: min power is $\min(0,1) = 0$ (7 is not common) So, $$\text{HCF} = 2^4 \times 3^2 \times 5^1$$ **Final answers:** - LCM: $2^5 \times 3^8 \times 5 \times 7$ - HCF: $2^4 \times 3^2 \times 5$