What is the least common multiple (LCM) of 30 and 54?

Steps to Solve

1. Prime Factorization of 30

First, we need to find the prime factors of 30. We can do this by dividing 30 by the smallest prime numbers until we reach 1.

$$ 30 \div 2 = 15 $$

$$ 15 \div 3 = 5 $$

$$ 5 \div 5 = 1 $$

Thus, the prime factorization of 30 is:

$$ 30 = 2^1 \times 3^1 \times 5^1 $$

2. Prime Factorization of 54

Next, we find the prime factors of 54 in the same way.

$$ 54 \div 2 = 27 $$

$$ 27 \div 3 = 9 $$

$$ 9 \div 3 = 3 $$

$$ 3 \div 3 = 1 $$

The prime factorization of 54 is:

$$ 54 = 2^1 \times 3^3 $$

3. Determine the LCM

To find the LCM, we take the highest power of each prime factor from both factorizations:

• For prime factor $2$: the highest power is $2^1$
• For prime factor $3$: the highest power is $3^3$
• For prime factor $5$: the highest power is $5^1$

Now, we can write the LCM as:

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

4. Calculate the LCM

Finally, we calculate the LCM:

$$ LCM = 2^1 \times 27 \times 5 $$

First, calculate $2 \times 27$:

$$ 2 \times 27 = 54 $$

Then multiply by 5:

$$ 54 \times 5 = 270 $$

Thus, the LCM of 30 and 54 is 270.