What is the least common multiple of 24 and 30?

Steps to Solve

1. 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:

   • $30 = 2^1 \times 3^1 \times 5^1$

2. 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)

3. Multiply the highest powers together

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

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

4. Calculate the result

   Finally, we compute the product:

   $$ \text{LCM} = 8 \times 3 \times 5 $$

   • First, calculate $8 \times 3 = 24$
   • Then calculate $24 \times 5 = 120$