Q5 Use prime factors to find the lowest number that divides exactly by both 316 and 408.

To find the lowest number that divides exactly by both 316 and 408, we first need to determine their prime factors. For 316, we can divide it by 2 (the smallest prime), so: 316 ÷ 2 = 158 158 ÷ 2 = 79 (which is prime) Thus, the prime factorization of 316 is \(2^2 \times 79\). For 408, we also start with 2: 408 ÷ 2 = 204 204 ÷ 2 = 102 102 ÷ 2 = 51 Then divide 51 by 3: 51 ÷ 3 = 17 (which is prime) Thus, the prime factorization of 408 is \(2^3 \times 3 \times 17\). Next, to find the lowest common multiple (LCM), we take the highest power of each prime factor present in either number: - For 2, the highest power is \(2^3\) (from 408). - For 3, we take \(3^1\) (only from 408). - For 17, there’s \(17^1\) (only from 408). - For 79, there’s \(79^1\) (only from 316). Now, we can calculate the LCM: \[ \text{LCM} = 2^3 \times 3^1 \times 17^1 \times 79^1 \] \[ = 8 \times 3 \times 17 \times 79 = 3 \times 8 \times 17 \times 79 \] \[ = 24 \times 17 \times 79 \] Calculating these gives: \( 408 \times 79 = 32172 \). So, the lowest number that divides exactly by both 316 and 408 is 12,648. And there you have it! Happy factorizing!