What is the least common multiple (LCM) of 9 and 15?
The correct answer and explanation is:
The least common multiple (LCM) of 9 and 15 is 45.
Explanation:
The LCM of two numbers is the smallest positive integer that is divisible by both of them. To find the LCM of 9 and 15, we can use the following steps:
- Find the prime factorization of both numbers:
- The prime factorization of 9 is 9=3×39 = 3 \times 3, or 323^2.
- The prime factorization of 15 is 15=3×515 = 3 \times 5.
- Identify the common and unique prime factors:
- Both 9 and 15 share the prime factor 3. However, 9 has two 3s (i.e., 323^2), while 15 only has one (i.e., 313^1).
- 15 has the additional prime factor 5, which does not appear in the prime factorization of 9.
- Take the highest power of each prime factor:
- For the prime factor 3, take the higher power, which is 323^2 (from 9).
- For the prime factor 5, take 515^1 (from 15).
- Multiply these highest powers together: LCM=32×51=9×5=45.LCM = 3^2 \times 5^1 = 9 \times 5 = 45.
Thus, the least common multiple of 9 and 15 is 45. This means that 45 is the smallest number that both 9 and 15 divide evenly into.
In conclusion, the LCM is an essential concept when dealing with multiples of numbers. It helps to find common timings or intervals where patterns or events overlap. The process can be generalized for more than two numbers as well.