The least common multiple of 3, 4, 6, and 8 is
A) 96.
B) 24.
C) 72.
D) 8.
Answer (2)
Find the prime factorization of each number: 3 = 3 , 4 = 2 2 , 6 = 2 × 3 , 8 = 2 3 .
Identify the highest power of each prime factor: 2 3 and 3 1 .
Multiply the highest powers together: 2 3 × 3 = 8 × 3 = 24 .
The least common multiple of 3, 4, 6, and 8 is 24 .
Explanation
Understanding the Problem We are asked to find the least common multiple (LCM) of the numbers 3, 4, 6, and 8. The LCM is the smallest positive integer that is divisible by all the given numbers.
Prime Factorization To find the LCM, we can use the prime factorization method. First, we find the prime factorization of each number:
3 = 3 4 = 2 2 6 = 2 × 3 8 = 2 3
Identifying Highest Powers Next, we identify the highest power of each prime factor present in the factorizations. The prime factors are 2 and 3. The highest power of 2 is 2 3 = 8 , and the highest power of 3 is 3 1 = 3 .
Calculating the LCM Now, we multiply these highest powers together to find the LCM:
LCM(3, 4, 6, 8) = 2 3 × 3 = 8 × 3 = 24
Final Answer Therefore, the least common multiple of 3, 4, 6, and 8 is 24.
Examples
Understanding the least common multiple is very useful in everyday life. For example, imagine you are planning a party and want to buy hot dogs and buns. Hot dogs come in packs of 6, and buns come in packs of 8. To ensure that you have an equal number of hot dogs and buns, you need to find the least common multiple of 6 and 8, which is 24. This means you need to buy 4 packs of hot dogs (4 * 6 = 24) and 3 packs of buns (3 * 8 = 24). This ensures that you have exactly the same number of hot dogs and buns, minimizing waste and making party planning easier!
The least common multiple (LCM) of 3, 4, 6, and 8 is 24. This is calculated by finding the prime factorization of each number, identifying the highest powers of each prime, and multiplying them together. Therefore, the correct answer is B) 24.
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