1. Using prime factorization, determine the least common multiple of 10, 12, 14

Step 1 ：First, we need to find the least common multiple (LCM) of 10, 12, and 14. The LCM of a set of numbers is the smallest number that is a multiple of each of the numbers in the set.

Step 2 ：One way to find the LCM is to use prime factorization. This involves breaking each number down into its prime factors, then multiplying each factor the greatest number of times it occurs in either number.

Step 3 ：Let's find the prime factorization of each number: \(10 = 2 * 5\), \(12 = 2^2 * 3\), and \(14 = 2 * 7\).

Step 4 ：Next, we take each prime factor and multiply it the greatest number of times it occurs in any of the factorizations. In this case, we have: 2 occurs twice in the factorization of 12, so we include it twice in our LCM. 3 occurs once in the factorization of 12, so we include it once in our LCM. 5 occurs once in the factorization of 10, so we include it once in our LCM. 7 occurs once in the factorization of 14, so we include it once in our LCM.

Step 5 ：So, the LCM of 10, 12, and 14 is \(2^2 * 3 * 5 * 7\).

Step 6 ：Final Answer: The least common multiple of 10, 12, and 14 is \(\boxed{420}\).