least common multiple of 24 and 42
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 24 and 42. To find the LCM, we will identify the prime factors of each number and then determine the smallest multiple that both can share.
Answer
The LCM of 24 and 42 is $168$.
Answer for screen readers
The least common multiple (LCM) of 24 and 42 is $168$.
Steps to Solve
- Find the prime factors of 24
To determine the LCM, we first find the prime factorization of 24.
The prime factorization of 24 is: $$ 24 = 2^3 \times 3^1 $$
- Find the prime factors of 42
Next, we find the prime factorization of 42.
The prime factorization of 42 is: $$ 42 = 2^1 \times 3^1 \times 7^1 $$
- Determine the highest powers of each prime factor
We take the highest power for each prime factor present in both factorizations:
- For the prime factor 2: The highest power is $2^3$ (from 24).
- For the prime factor 3: The highest power is $3^1$ (common to both).
- For the prime factor 7: The highest power is $7^1$ (from 42).
- Calculate the LCM
Now, we multiply these highest powers together to find the LCM:
$$ LCM = 2^3 \times 3^1 \times 7^1 $$
Calculating this gives:
$$ LCM = 8 \times 3 \times 7 = 24 \times 7 = 168 $$
The least common multiple (LCM) of 24 and 42 is $168$.
More Information
The least common multiple is the smallest number that is a multiple of both given numbers. It is useful in various applications, such as finding common denominators in fractions.
Tips
- A common mistake is forgetting to take the highest power of each prime factor. Always ensure you consider the highest exponent when finding the LCM.
- Another mistake is assuming that the LCM is just the product of the numbers without considering their prime factors, which can lead to a much larger number than necessary.
