least common multiple of 30 and 54

Understand the Problem

The question is asking for the least common multiple (LCM) of the numbers 30 and 54. This involves finding the smallest positive integer that is divisible by both of these numbers.

Answer

The least common multiple of 30 and 54 is $270$.

Steps to Solve

Find the Prime Factorization of Each Number

First, we need to find the prime factors of each number.

For 30:

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

For 54:

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

Identify the Highest Powers of Each Prime Factor

Next, we take each prime factor from both factorizations and find the highest power that appears.

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

Multiply the Highest Powers Together

Now, we multiply these highest powers to find the LCM.

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

Calculate the LCM

Calculating the product step-by-step:

Calculate $2^1 = 2$
Calculate $3^3 = 27$
Calculate $5^1 = 5$

Then, multiply these results: $$ LCM = 2 \times 27 \times 5 $$

Now, let's multiply:

First, $2 \times 27 = 54$
Then, $54 \times 5 = 270$

Therefore, the LCM of 30 and 54 is 270.

The least common multiple of 30 and 54 is $270$.

More Information

The least common multiple (LCM) is useful in various applications, including adding fractions with different denominators and finding common periods in arithmetic sequences. The LCM can also be calculated through the relationship with the greatest common divisor (GCD) using the formula: $$ LCM(a, b) = \frac{|a \times b|}{GCD(a, b)} $$

Tips

Not finding the correct prime factorization can lead to an incorrect LCM.
Forgetting to consider all prime factors when multiplying the highest powers.

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