EXERCISE 16 Use prime factorisation to determine the HCF of the following numbers; (1) 8 and 16 (2) 9 and 27 (3) 18 and \( _{42} \) (4) 54 and 98 (5) 90 and 75 (7) 84 and 126 (8) 300 and 250 \( (10)^{*} 2250,7560 \) and 10500

Certainly! Let's determine the Highest Common Factor (HCF) of each pair of numbers using **prime factorization**. The HCF is the largest number that divides each of the numbers without leaving a remainder. --- ### **(1) HCF of 8 and 16** - **Prime Factorization:** - \(8 = 2^3\) - \(16 = 2^4\) - **HCF Calculation:** - **Common Prime Factor:** 2 - **Lowest Power:** \(2^3\) - **HCF:** \(2^3 = 8\) --- ### **(2) HCF of 9 and 27** - **Prime Factorization:** - \(9 = 3^2\) - \(27 = 3^3\) - **HCF Calculation:** - **Common Prime Factor:** 3 - **Lowest Power:** \(3^2\) - **HCF:** \(3^2 = 9\) --- ### **(3) HCF of 18 and 42** *(Assuming the second number is 42 based on the notation)* - **Prime Factorization:** - \(18 = 2 \times 3^2\) - \(42 = 2 \times 3 \times 7\) - **HCF Calculation:** - **Common Prime Factors:** 2 and 3 - **Lowest Powers:** \(2^1\) and \(3^1\) - **HCF:** \(2 \times 3 = 6\) --- ### **(4) HCF of 54 and 98** - **Prime Factorization:** - \(54 = 2 \times 3^3\) - \(98 = 2 \times 7^2\) - **HCF Calculation:** - **Common Prime Factor:** 2 - **Lowest Power:** \(2^1\) - **HCF:** \(2\) --- ### **(5) HCF of 90 and 75** - **Prime Factorization:** - \(90 = 2 \times 3^2 \times 5\) - \(75 = 3 \times 5^2\) - **HCF Calculation:** - **Common Prime Factors:** 3 and 5 - **Lowest Powers:** \(3^1\) and \(5^1\) - **HCF:** \(3 \times 5 = 15\) --- ### **(7) HCF of 84 and 126** - **Prime Factorization:** - \(84 = 2^2 \times 3 \times 7\) - \(126 = 2 \times 3^2 \times 7\) - **HCF Calculation:** - **Common Prime Factors:** 2, 3, and 7 - **Lowest Powers:** \(2^1\), \(3^1\), and \(7^1\) - **HCF:** \(2 \times 3 \times 7 = 42\) --- ### **(8) HCF of 300 and 250** - **Prime Factorization:** - \(300 = 2^2 \times 3 \times 5^2\) - \(250 = 2 \times 5^3\) - **HCF Calculation:** - **Common Prime Factors:** 2 and 5 - **Lowest Powers:** \(2^1\) and \(5^2\) - **HCF:** \(2 \times 5^2 = 2 \times 25 = 50\) --- ### **(10) HCF of 2250, 7560, and 10500** - **Prime Factorization:** - \(2250 = 2 \times 3^2 \times 5^3\) - \(7560 = 2^3 \times 3^3 \times 5 \times 7\) - \(10500 = 2^2 \times 3 \times 5^3 \times 7\) - **HCF Calculation:** - **Common Prime Factors:** 2, 3, and 5 - **Lowest Powers:** - For 2: \(2^1\) (from 2250) - For 3: \(3^1\) (from 10500) - For 5: \(5^1\) (from 7560) - **HCF:** \(2^1 \times 3^1 \times 5^1 = 2 \times 3 \times 5 = 30\) --- ### **Summary of HCFs:** 1. **8 and 16:** **8** 2. **9 and 27:** **9** 3. **18 and 42:** **6** 4. **54 and 98:** **2** 5. **90 and 75:** **15** 7. **84 and 126:** **42** 8. **300 and 250:** **50** 10. **2250, 7560, and 10500:** **30** --- Feel free to reach out if you have any more questions or need further clarifications!