What is the least common multiple of 24 and 30?

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 24 and 30. To solve this, we can start by finding the multiples of each number and identifying the smallest multiple they share.

Answer

The LCM of 24 and 30 is $120$.

Steps to Solve

Find the prime factorization of each number

First, we will find the prime factors of 24 and 30.

For 24:
- $24 = 2^3 \times 3^1$
For 30:
- $30 = 2^1 \times 3^1 \times 5^1$
Identify the highest power of each prime number

Next, we will look at all the prime factors from both factorizations and take the highest power for each prime.
- For the prime number 2: highest power is $2^3$ (from 24)
- For the prime number 3: highest power is $3^1$ (both have the same)
- For the prime number 5: highest power is $5^1$ (only from 30)
Multiply the highest powers together

Now, we will calculate the LCM by multiplying the highest powers of all prime factors together.

$$ \text{LCM} = 2^3 \times 3^1 \times 5^1 $$
Calculate the result

Finally, we compute the product:

$$ \text{LCM} = 8 \times 3 \times 5 $$
- First, calculate $8 \times 3 = 24$
- Then calculate $24 \times 5 = 120$

The least common multiple (LCM) of 24 and 30 is 120.

More Information

The least common multiple (LCM) is important in various applications such as adding fractions or solving problems involving periodic events. The method of prime factorization is often the most effective way to find the LCM for multiple numbers.

Tips

A common mistake is to list out the multiples of each number without considering the efficiency of prime factorization. This can lead to unnecessary calculations and time consumption.
Another mistake is overlooking one of the prime factors or their highest powers when calculating the LCM.

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