.1 Write 56,84 and 104 each as a product of their prime factors, showing all your working. 2 Give the HCF of 56,84 and 104 . 1 Write 12 and 18 each as a product of their prime factors. 2 Hence, give the lowest common multiple of 12 and 18 , i.e. the LCM: .3 Now write 70 as a product of its prime factors, (3) and hence find the LCM of 12,18 and 70 .

To find the prime factorization of these numbers, let’s break them down step by step. 1. **For 56**: - 56 ÷ 2 = 28 - 28 ÷ 2 = 14 - 14 ÷ 2 = 7 (7 is prime) - Therefore, the prime factorization of 56 is \(2^3 \times 7\). 2. **For 84**: - 84 ÷ 2 = 42 - 42 ÷ 2 = 21 - 21 ÷ 3 = 7 (7 is prime) - Therefore, the prime factorization of 84 is \(2^2 \times 3 \times 7\). 3. **For 104**: - 104 ÷ 2 = 52 - 52 ÷ 2 = 26 - 26 ÷ 2 = 13 (13 is prime) - Therefore, the prime factorization of 104 is \(2^3 \times 13\). **Now, finding the HCF of 56, 84, and 104**: - The common prime factors are \(2^2\) (minimum power of 2 in all factorizations). - Thus, the HCF = \(2^2 = 4\). --- 1. **For 12**: - 12 ÷ 2 = 6 - 6 ÷ 2 = 3 (3 is prime) - Therefore, the prime factorization of 12 is \(2^2 \times 3\). 2. **For 18**: - 18 ÷ 2 = 9 - 9 ÷ 3 = 3 (3 is prime) - Therefore, the prime factorization of 18 is \(2 \times 3^2\). **Now, for the LCM of 12 and 18**: - The LCM is found by taking the highest power of each prime number present: - \(2^2\) from 12 - \(3^2\) from 18 - Therefore, the LCM = \(2^2 \times 3^2 = 4 \times 9 = 36\). --- 3. **For 70**: - 70 ÷ 2 = 35 - 35 ÷ 5 = 7 (7 is prime) - Therefore, the prime factorization of 70 is \(2 \times 5 \times 7\). **Now, finding the LCM of 12, 18, and 70**: - For LCM, we take the highest power of each prime: - \(2^2\) from 12 - \(3^2\) from 18 - \(5^1\) from 70 - \(7^1\) from both 56 and 84 - Therefore, the LCM = \(2^2 \times 3^2 \times 5 \times 7 = 4 \times 9 \times 5 \times 7\). Calculating this gives us \(36 \times 5 = 180\) and \(180 \times 7 = 1260\). - Thus, the LCM of 12, 18, and 70 is 1260.