What is the least common multiple of 9, 10, and 12?

The least common multiple (LCM) of the numbers 9, 10, and 12 is 180.

To find the LCM, we can use the prime factorization method:

• 9 can be factored into 3².
• 10 can be factored into 2¹ × 5¹.
• 12 can be factored into 2² × 3¹.

Now, we take the highest power of each prime number that appears in these factorizations:

• For the prime number 2, the highest power is 2² (from 12).
• For the prime number 3, the highest power is 3² (from 9).
• For the prime number 5, the highest power is 5¹ (from 10).

Now we can multiply these together to find the LCM:

LCM = 2² × 3² × 5¹ = 4 × 9 × 5 = 180

Therefore, the least common multiple of 9, 10, and 12 is 180.