What is the least common multiple of 9 and 24?
Understand the Problem
The question is asking for the least common multiple (LCM) of two numbers, 9 and 24. To find the LCM, we can determine the multiples of both numbers and identify the smallest multiple they share.
Answer
The least common multiple of 9 and 24 is $72$.
Answer for screen readers
The least common multiple (LCM) of 9 and 24 is $72$.
Steps to Solve
- Determine the prime factorization of each number
Start by finding the prime factors of both 9 and 24.
- For 9: The prime factorization is $9 = 3^2$.
- For 24: The prime factorization is $24 = 2^3 \times 3^1$.
- Identify the highest power of each prime factor
Next, look at the prime factorization and identify the highest powers of each prime that appear in the factorizations of both numbers.
- The primes are 2 and 3.
- For 2: The highest power is $2^3$ (from 24).
- For 3: The highest power is $3^2$ (from 9).
- Multiply the highest powers together
Now, we multiply the highest powers of all prime factors together to find the LCM. $$ \text{LCM} = 2^3 \times 3^2 $$
- Calculate the final result
Now, calculate the product:
- $2^3 = 8$
- $3^2 = 9$
- Therefore, the LCM is $8 \times 9 = 72$.
The least common multiple (LCM) of 9 and 24 is $72$.
More Information
The least common multiple (LCM) is the smallest number that is a multiple of both numbers. It is useful in solving problems involving fractions, as it helps in finding a common denominator.
Tips
- A common mistake is to simply multiply the two numbers together. However, the LCM is not always the product of the two numbers. It requires finding the greatest power of all prime factors involved.
- Forgetting to consider all prime factors can also lead to an incorrect LCM. Make sure to include all primes from both factorizations.
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