What is the least common multiple of 30 and 8?

Steps to Solve

1. Identify the Prime Factors of Each Number

First, we'll find the prime factors of 30 and 8.

• The prime factors of 30 are: $$ 30 = 2 \times 3 \times 5 $$

• The prime factors of 8 are: $$ 8 = 2^3 $$

2. List All Unique Prime Factors

Next, we need to list all unique prime factors from both numbers.

• The unique prime factors are: $$ 2, 3, 5 $$

3. Determine the Highest Power of Each Prime Factor

Now we find the highest power of each prime factor present in either number:

• For the prime factor 2, the highest power is (2^3) (from 8).
• For the prime factor 3, the highest power is (3^1) (from 30).
• For the prime factor 5, the highest power is (5^1) (from 30).

4. Calculate the LCM

Finally, we multiply the highest powers of all prime factors to get the least common multiple:

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

Calculating this gives:

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

5. Finalize the Multiplication

Now we perform the multiplication step-by-step:

First, calculate (8 \times 3):

$$ 8 \times 3 = 24 $$

Then multiply it by 5:

$$ 24 \times 5 = 120 $$

Thus,

$$ \text{LCM}(30, 8) = 120 $$