1. **Stating the problem:** We have three numbers with HCF (Highest Common Factor) 8 and LCM (Least Common Multiple) 2520. One number is a multiple of 9, another a multiple of 5, and another a multiple of 7. We need to find the smaller, larger, and the other number. 2. **Formula and rules:** For any set of numbers, the product of the numbers equals the product of their HCF and LCM times some factor depending on their pairwise relationships. Since the HCF is 8, each number can be written as $8a$, $8b$, and $8c$ where $a,b,c$ are integers with HCF 1. 3. **Expressing the problem:** Given the numbers are $8a$, $8b$, $8c$ with HCF$(a,b,c)=1$ and LCM$(8a,8b,8c)=2520$. Since LCM$(8a,8b,8c) = 8 \times \text{LCM}(a,b,c) = 2520$, then $$\text{LCM}(a,b,c) = \frac{2520}{8} = 315.$$ 4. **Prime factorization:** Factorize 315: $$315 = 3^2 \times 5 \times 7.$$ 5. **Conditions on $a,b,c$:** - One number is multiple of 9, so one of $a,b,c$ is multiple of 9 ($3^2$). - One number is multiple of 5. - One number is multiple of 7. Since $a,b,c$ have HCF 1, they share no common prime factors. 6. **Assigning factors:** Let: - $a = 9 = 3^2$ - $b = 5$ - $c = 7$ Check LCM$(a,b,c)$: $$\text{LCM}(9,5,7) = 3^2 \times 5 \times 7 = 315,$$ which matches the required LCM. 7. **Find the original numbers:** $$8a = 8 \times 9 = 72,$$ $$8b = 8 \times 5 = 40,$$ $$8c = 8 \times 7 = 56.$$ 8. **Identify smaller, larger, and other numbers:** - Smaller number: $40$ - Larger number: $72$ - Other number: $56$ 9. **Answer to question 12:** Number of different pairs with HCF 5 and LCM 1050. Use formula: $$\text{Product} = \text{HCF} \times \text{LCM} = 5 \times 1050 = 5250.$$ If two numbers are $5x$ and $5y$ with HCF$(x,y)=1$, then $$5x \times 5y = 5250 \Rightarrow 25xy = 5250 \Rightarrow xy = 210.$$ Number of pairs $(x,y)$ with $xy=210$ and HCF$(x,y)=1$ equals number of pairs of coprime factors of 210. Factorize 210: $$210 = 2 \times 3 \times 5 \times 7.$$ Number of divisors is $2^4=16$. Number of pairs of coprime factors is $2^{4-1} = 8$. So, there are 8 different pairs. **Final answers:** - Smaller number: $40$ - Larger number: $72$ - Other number: $56$ - Number of pairs with HCF 5 and LCM 1050: 8