lcm 15 and 27

Steps to Solve

1. Identify the multiples of each number

First, let's list the multiples of 15 and 27.

The multiples of 15 are:
$15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225, ...$

The multiples of 27 are:
$27, 54, 81, 108, 135, 162, 189, 216, 243, ...$

2. Find the smallest common multiple

Next, we check both lists for the smallest number that appears in both lists.

From our lists:

• The multiples of 15 include: $15, 30, 45, \ldots, 135, \ldots$
• The multiples of 27 include: $27, 54, \ldots, 135, \ldots$

The first common multiple we see is $135$.

3. Verify that it's the least common multiple (LCM)

To confirm that 135 is indeed the LCM, we can check the formulas for LCM:
The LCM of two numbers can also be found using the formula:
$$ LCM(a, b) = \frac{|a \times b|}{GCD(a, b)} $$
where GCD is the greatest common divisor.

4. Calculate the GCD of 15 and 27

To find the GCD, we can use the prime factorization method:

• The prime factors of 15 are $3 \times 5$.
• The prime factors of 27 are $3^3$.

The GCD is the product of the lowest powers of the common factors:
The only common factor is $3$, so $GCD(15, 27) = 3$.

5. Calculate the LCM using the GCD

Now we can calculate the LCM using the numbers we have:
$$ LCM(15, 27) = \frac{|15 \times 27|}{GCD(15, 27)} = \frac{405}{3} = 135 $$