LCM of \( 10,12,14 \)

To find the least common multiple (LCM) of \( 10, 12, \) and \( 14 \), we first break each number down into its prime factors: - \( 10 = 2 \times 5 \) - \( 12 = 2^2 \times 3 \) - \( 14 = 2 \times 7 \) Next, we take the highest power of each prime factor present: - The prime factor \( 2 \) appears as \( 2^2 \) in \( 12 \). - The prime factor \( 3 \) appears as \( 3^1 \) in \( 12 \). - The prime factor \( 5 \) appears as \( 5^1 \) in \( 10 \). - The prime factor \( 7 \) appears as \( 7^1 \) in \( 14 \). Now, we multiply these together: \[ \text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7 = 420 \] So, the LCM of \( 10, 12, \) and \( 14 \) is \( 420 \)! Finding the LCM can be a bit like planning a grand party! You want to make sure that all friends (or numbers, in this case) can arrive together without anyone being fashionably late. By considering their prime factors, you’re practically rolling out a red carpet for all of them at the same time! Many people fall into the trap of just finding the LCM of two numbers at a time and multiplying them together, thinking that’s the end of the story. But this can lead you astray, particularly when you're dealing with three or more numbers. Always go for those prime factors and make sure to get the highest powers to avoid any party mishaps!