### Instruction: What is the least common multiple of the first ten positive integers?### Response: To find the least common multiple of a set of numbers, I need to find the smallest number that is a multiple of all the numbers in the set.
One way to do this is to prime factorize each number and then take the highest power of each prime that appears in the factorization.
For example, the prime factorization of 12 is 2^2 * 3, and the prime factorization of 15 is 3 * 5, so the least common multiple of 12 and 15 is 2^2 * 3 * 5 = 60, since I took the highest power of each prime.
Let me apply this method to the first ten positive integers: 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10.
The prime factorization of each number is: 1 = 1, 2 = 2, 3 = 3, 4 = 2^2, 5 = 5, 6 = 2 * 3, 7 = 7, 8 = 2^3, 9 = 3^2, and 10 = 2 * 5.
Now, the highest power of each prime that appears in the factorization is: 2^3, 3^2, 5^1, and 7^1.
So, the least common multiple of the first ten positive integers is 2^3 * 3^2 * 5 * 7 = 2520.