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

1. Factorize each number into its prime factors. - For \(316\): \[ 316 \div 2 = 158,\quad 158 \div 2 = 79. \] Since \(79\) is prime, we have \[ 316 = 2^2 \times 79. \] - For \(408\): \[ 408 \div 2 = 204,\quad 204 \div 2 = 102,\quad 102 \div 2 = 51. \] Now, factorize \(51\): \[ 51 = 3 \times 17. \] So, \[ 408 = 2^3 \times 3 \times 17. \] 2. To find the lowest number (least common multiple, LCM) that is exactly divisible by both numbers, take each prime factor raised to the highest power that appears in either factorization. - For the prime \(2\): Maximum power is \(2^3\) (from \(408\)). - For the prime \(3\): Maximum power is \(3^1\) (from \(408\)). - For the prime \(17\): Maximum power is \(17^1\) (from \(408\)). - For the prime \(79\): Maximum power is \(79^1\) (from \(316\)). Therefore, the LCM is \[ \text{LCM} = 2^3 \times 3 \times 17 \times 79. \] 3. Compute the LCM step by step: - First, calculate \(2^3 \times 3\): \[ 2^3 \times 3 = 8 \times 3 = 24. \] - Next, multiply by \(17\): \[ 24 \times 17 = 408. \] - Finally, multiply by \(79\): \[ 408 \times 79. \] You can calculate this as: \[ 408 \times 80 = 32640,\quad \text{then subtract }408: \quad 32640 - 408 = 32232. \] Hence, the lowest number that divides exactly by both \(316\) and \(408\) is \[ 32232. \]