AA oclow. Draw the prime factor tree for 105 and use it to find the lowest common multiple (LCM) of 105 and 462. Watch video sparxmaths.uk Answer

# Explanation:<br /><br />## Step 1:<br />First, we need to find the prime factorization of 105. We start by dividing 105 by the smallest prime number, which is 2. Since 105 is not divisible by 2, we try the next prime number, which is 3. It's also not divisible by 3. The next prime number is 5, and 105 is divisible by 5. So, we divide 105 by 5 to get 21. <br /><br />### \(105 = 5 \times 21\)<br /><br />## Step 2:<br />We continue the process with 21. The smallest prime number that 21 is divisible by is 3. So, we divide 21 by 3 to get 7. <br /><br />### \(21 = 3 \times 7\)<br /><br />## Step 3:<br />Now, we have 7 which is a prime number itself. So, the prime factorization of 105 is 5, 3, and 7.<br /><br />### \(105 = 5 \times 3 \times 7\)<br /><br />## Step 4:<br />Next, we need to find the prime factorization of 462. We start by dividing 462 by the smallest prime number, which is 2. 462 is divisible by 2, so we divide 462 by 2 to get 231.<br /><br />### \(462 = 2 \times 231\)<br /><br />## Step 5:<br />We continue the process with 231. The smallest prime number that 231 is divisible by is 3. So, we divide 231 by 3 to get 77.<br /><br />### \(231 = 3 \times 77\)<br /><br />## Step 6:<br />We continue the process with 77. The smallest prime number that 77 is divisible by is 7. So, we divide 77 by 7 to get 11.<br /><br />### \(77 = 7 \times 11\)<br /><br />## Step 7:<br />Now, we have 11 which is a prime number itself. So, the prime factorization of 462 is 2, 3, 7, and 11.<br /><br />### \(462 = 2 \times 3 \times 7 \times 11\)<br /><br />## Step 8:<br />To find the least common multiple (LCM) of 105 and 462, we take each prime factor the greatest number of times it appears in either factorization. So, the LCM is \(2 \times 3 \times 5 \times 7 \times 11\).<br /><br />### \(LCM(105, 462) = 2 \times 3 \times 5 \times 7 \times 11\)<br /><br /># Answer: <br /><br />The prime factorization of 105 is \(5 \times 3 \times 7\) and the prime factorization of 462 is \(2 \times 3 \times 7 \times 11\). The least common multiple (LCM) of 105 and 462 is \(2 \times 3 \times 5 \times 7 \times 11\).