What is the least common multiple of 24 and 48?
Understand the Problem
The question is asking for the least common multiple (LCM) of the numbers 24 and 48. To find the LCM, we look for the smallest multiple that is common to both numbers. Typically, this can be calculated by identifying the prime factors of each number and using them to determine the LCM.
Answer
$48$
Answer for screen readers
The least common multiple (LCM) of 24 and 48 is $48$.
Steps to Solve
- Find the prime factorization of 24 First, we need to find the prime factors of 24. We can do this by dividing it by the smallest prime numbers.
$$ 24 = 2 \times 12 $$ $$ 12 = 2 \times 6 $$ $$ 6 = 2 \times 3 $$ So, the full prime factorization of 24 is: $$ 24 = 2^3 \times 3^1 $$
- Find the prime factorization of 48 Next, we will do the same for 48.
$$ 48 = 2 \times 24 $$ Since we already know the prime factorization of 24, we can write: $$ 48 = 2 \times (2^3 \times 3^1) $$ This gives us: $$ 48 = 2^4 \times 3^1 $$
- Determine the LCM using the prime factors To find the LCM, we take the highest power of each prime factor from both factorizations.
- For the prime factor 2: $2^4$ (from 48)
- For the prime factor 3: $3^1$ (common in both)
Thus, the LCM can be computed as follows:
$$ \text{LCM} = 2^4 \times 3^1 $$
- Calculate the LCM Now we can calculate it:
$$ \text{LCM} = 16 \times 3 = 48 $$
The least common multiple (LCM) of 24 and 48 is $48$.
More Information
The LCM of two numbers is useful in situations where you need to add or subtract fractions with different denominators or to find intervals in repeating events. The LCM allows you to find a common point where these intervals align.
Tips
- A common mistake is to assume that the LCM is the largest number. In fact, it is the smallest common multiple of the two numbers.
- Another mistake is not considering all prime factors or taking the minimum instead of the highest power when calculating the LCM.
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