What is the least common multiple of 24 and 9?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 24 and 9. To find the LCM, we look for the smallest number that is a multiple of both of these numbers. The approach involves determining the prime factors of each number, finding the highest power of each prime factor, and then multiplying these together.
Answer
The LCM of 24 and 9 is 72.
Answer for screen readers
The least common multiple (LCM) of 24 and 9 is 72.
Steps to Solve
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Find the prime factors of each number
Start by determining the prime factorization of 24 and 9.
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The prime factors of 24 are: $$ 24 = 2^3 \times 3^1 $$
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The prime factors of 9 are: $$ 9 = 3^2 $$
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Determine the highest power of each prime factor
Identify the prime factors from both factorizations and take the highest power for each.
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For the prime factor 2: the highest power is $2^3$ (from 24).
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For the prime factor 3: the highest power is $3^2$ (from 9).
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Multiply the highest powers to find the LCM
Multiply the values of the highest powers obtained in the previous step.
- The LCM is calculated as follows: $$ \text{LCM} = 2^3 \times 3^2 $$
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Calculate the result
Now calculate the values:
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First, compute $2^3 = 8$ and $3^2 = 9$.
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Finally, multiply these results: $$ \text{LCM} = 8 \times 9 = 72 $$
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The least common multiple (LCM) of 24 and 9 is 72.
More Information
The least common multiple is useful in problems that require finding common denominators or when adding fractions. In this case, 72 is the smallest number that both 24 and 9 can divide without leaving a remainder.
Tips
- Forgetting to factor each number correctly. Make sure to include all prime factors.
- Confusing LCM with greatest common divisor (GCD). LCM finds a common multiple, while GCD finds a common divisor. Confirm the problem is asking for LCM.
