LCM of 54 and 30

Steps to Solve

1. Finding the Prime Factorization First, we need to find the prime factorization of both numbers:

For 54:
54 can be divided by 2 to give 27.
Then, 27 can be divided by 3 to give 9.
Next, 9 can be divided by 3 to give 3.
Finally, 3 is a prime number.

The prime factorization of 54 is:
$$ 54 = 2^1 \times 3^3 $$

For 30:
30 can be divided by 2 to give 15.
Then, 15 can be divided by 3 to give 5.
Finally, 5 is a prime number.

The prime factorization of 30 is:
$$ 30 = 2^1 \times 3^1 \times 5^1 $$

2. Taking the Highest Power of Each Prime Factor Next, we take the highest power of each prime factor from both numbers:

• For the prime factor 2, the highest power is $2^1$ (from both).
• For the prime factor 3, the highest power is $3^3$ (from 54).
• For the prime factor 5, the highest power is $5^1$ (from 30).

3. Calculating the LCM Now we can calculate the LCM by multiplying these highest powers together:

$$ LCM = 2^1 \times 3^3 \times 5^1 $$

4. Performing the Calculation Calculating this step-by-step:

• First calculate $3^3$: $$ 3^3 = 27 $$

• Now multiply: $$ LCM = 2^1 \times 27 \times 5^1 = 2 \times 27 \times 5 $$

Calculating $2 \times 27 = 54$ and then $54 \times 5 = 270$.

Thus, the least common multiple of 54 and 30 is 270.