What is the LCM of 6, 12, and 15?
Understand the Problem
The question is asking for the Least Common Multiple (LCM) of the numbers 6, 12, and 15. To find the LCM, we will determine the smallest number that is a multiple of all three given numbers.
Answer
$60$
Answer for screen readers
The Least Common Multiple (LCM) of 6, 12, and 15 is $60$.
Steps to Solve
- Find the prime factorization of each number
Start by determining the prime factorization of each number.
- For 6: $6 = 2 \times 3$
- For 12: $12 = 2^2 \times 3$
- For 15: $15 = 3 \times 5$
- Identify the highest power of each prime factor
Next, identify the highest power of each prime factor present in any of the factorizations.
- The highest power of 2: $2^2$ (from 12)
- The highest power of 3: $3^1$ (from 6, 12, and 15)
- The highest power of 5: $5^1$ (from 15)
- Multiply these highest powers together
Finally, multiply the highest powers of the prime factors together to get the LCM.
$$ \text{LCM} = 2^2 \times 3^1 \times 5^1 $$
Calculating this gives:
$$ \text{LCM} = 4 \times 3 \times 5 = 12 \times 5 = 60 $$
The Least Common Multiple (LCM) of 6, 12, and 15 is $60$.
More Information
The least common multiple of numbers is useful in various mathematical applications, including finding common denominators in fractions, scheduling events, and solving problems that involve periodicity.
Tips
- Forgetting to consider all prime factors: Ensure every prime factor is considered in its highest power.
- Confusing LCM with GCD (greatest common divisor): LCM is the smallest common multiple, while GCD is the largest common factor.
