What is the LCM of 9, 12, and 15?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 9, 12, and 15. To solve this, we need to find a number that is a multiple of each of these three numbers and is the smallest such number.
Answer
$180$
Answer for screen readers
The least common multiple (LCM) of 9, 12, and 15 is $180$.
Steps to Solve
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Find the Prime Factorizations
First, we need to find the prime factorization of each of the numbers.
- For 9: $9 = 3^2$
- For 12: $12 = 2^2 \times 3^1$
- For 15: $15 = 3^1 \times 5^1$
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Identify the Highest Powers of Each Prime Factor
Next, we will identify the highest powers of all prime factors present in the factorizations.
- The prime factors are: 2, 3, and 5.
- The highest powers are:
- For 2: $2^2$ (from 12)
- For 3: $3^2$ (from 9)
- For 5: $5^1$ (from 15)
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Calculate the LCM
To find the LCM, we multiply the highest powers of each prime factor together: $$ \text{LCM} = 2^2 \times 3^2 \times 5^1 $$ -
Perform the Multiplication
Now we can calculate the LCM step by step:
- First, calculate $2^2 = 4$.
- Then, calculate $3^2 = 9$.
- Now, multiply these results with 5: $$ \text{LCM} = 4 \times 9 \times 5 $$
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Final Calculation
Calculate $4 \times 9 = 36$, then multiply $36 \times 5 = 180$.
The least common multiple (LCM) of 9, 12, and 15 is $180$.
More Information
The least common multiple is the smallest number into which all the given numbers divide evenly. In this case, 180 is the smallest number that can be exactly divided by 9, 12, and 15. Knowing how to find the LCM is very useful in fraction calculations, scheduling problems, and many areas of number theory.
Tips
- Not recognizing the need for prime factorization: Some might attempt to list multiples instead. This can be inefficient and lead to errors.
- Neglecting to take the highest power of each prime factor: It's crucial to consider the maximum power when determining the LCM.
