Final answer:
The least common multiple of 9, 12, and 15 is calculated by finding the highest power of each prime that appears in their prime factorizations and multiplying those together, which results in 180.
Step-by-step explanation:
To find the least common multiple (LCM) of 9, 12, and 15, we must find the smallest number that all three numbers divide into without leaving a remainder. First, we'll find the prime factorization of each number:
- 9 = 3²
- 12 = 2² × 3
- 15 = 3 × 5
Next, we'll take each prime number that appears in the factorization the greatest number of times it appears in any of the factorizations:
- The highest power of 2 is 2² from the number 12.
- The highest power of 3 is 3² from the number 9.
- The highest power of 5 is 5 from the number 15.
Now, we multiply these together to get the LCM:
LCM = 2² × 3² × 5 = 4 × 9 × 5 = 36 × 5 = 180.
Therefore, the least common multiple of 9, 12, and 15 is 180.