Find the LCM of 15 and 9.
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 15 and 9. To solve it, we will find the multiples of both numbers and identify the smallest common multiple.
Answer
45
Answer for screen readers
The least common multiple of 15 and 9 is ( \text{LCM}(15, 9) = 45 ).
Steps to Solve
- List the multiples of each number
First, we will find the multiples of 15 and 9.
Multiples of 15:
15, 30, 45, 60, 75, 90, 105, 120, ...
Multiples of 9:
9, 18, 27, 36, 45, 54, 63, 72, 81, 90, ...
- Identify the common multiples
Next, we will identify the multiples that appear in both lists:
Common multiples of 15 and 9:
45, 90, ...
- Find the least common multiple (LCM)
Now, we will find the smallest number that is common in both lists:
The least common multiple is:
$$ \text{LCM}(15, 9) = 45 $$
- Confirm the result using prime factorization (optional)
To double-check, we can use prime factorization.
- For 15: $3 \times 5$
- For 9: $3^2$
Now, take the highest power of each prime number:
- Highest power of 3 is $3^2$
- Highest power of 5 is $5^1$
Now calculate the LCM:
$$ \text{LCM}(15, 9) = 3^2 \times 5^1 = 9 \times 5 = 45 $$
The least common multiple of 15 and 9 is ( \text{LCM}(15, 9) = 45 ).
More Information
The least common multiple (LCM) is used to find a common denominator in fractions or to solve problems involving timing events or repeating cycles. The LCM of 15 and 9 is also interesting in that it is the first number that both can evenly divide.
Tips
- Forgetting to list enough multiples or checking only a few. Be sure to list enough multiples to find the LCM.
- Confusing LCM with greatest common divisor (GCD). Remember, LCM is the smallest common multiple, while GCD is the largest common factor.
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