What is the least common multiple (LCM) of 6, 8, and 12?

To determine the least common multiple (LCM) of the numbers 6, 8, and 12, we first need to find their prime factorizations:

• The prime factorization of 6 is (2^1 \times 3^1).

• The prime factorization of 8 is (2^3).

• The prime factorization of 12 is (2^2 \times 3^1).

The LCM is found by taking the highest power of each prime that appears in the factorizations:

• For the prime number 2, the highest power is (2^3) (from 8).

• For the prime number 3, the highest power is (3^1) (from both 6 and 12).

Now, we combine these:

[

LCM = 2^3 \times 3^1 = 8 \times 3 = 24

]

Thus, the least common multiple of 6, 8, and 12 is 24. This answer makes sense because 24 is a multiple of each of the original numbers. It is the smallest number that can be evenly divided by 6, 8,