If two numbers are in the ratio 2:6 and their LCM is 180. Find the numbers 

To solve the problem where two numbers are in the ratio 2:6, and their Least Common Multiple (LCM) is 180, follow these steps:

### Step 1: Express the numbers in terms of a variable.
Let the numbers be \( 2x \) and \( 6x \), where \( x \) is a common factor.

### Step 2: Find the LCM of \( 2x \) and \( 6x \).
- The LCM of two numbers is found by taking the highest powers of all the prime factors that appear in the factorization of the numbers.
- The prime factorizations of \( 2x \) and \( 6x \) are:
  - \( 2x = 2 \times x \)
  - \( 6x = 2 \times 3 \times x \)

Thus, the LCM of \( 2x \) and \( 6x \) is:
\[
\text{LCM}(2x, 6x) = 2 \times 3 \times x = 6x
\]

### Step 3: Set the LCM equal to 180.
Since the LCM is given as 180, we can set:
\[
6x = 180
\]

### Step 4: Solve for \( x \).
\[
x = \frac{180}{6} = 30
\]

### Step 5: Find the two numbers.
Substitute \( x = 30 \) into the expressions for the numbers:
- First number: \( 2x = 2 \times 30 = 60 \)
- Second number: \( 6x = 6 \times 30 = 180 \)

Thus, the two numbers are **60** and **180**.

### Final Answer:
The two numbers are **60** and **180**.

Would you like further details or clarifications? Here are five related questions to explore further:

1. What is the greatest common divisor (GCD) of 60 and 180?
2. Can you find the LCM of other number pairs in a similar ratio?
3. How does changing the ratio affect the resulting numbers?
4. How can you verify the LCM using prime factorization?
5. Can you solve this problem using another method, like directly from the ratio?

**Tip:** To quickly find the LCM of two numbers, break them into their prime factors and choose the highest powers of all primes that appear.

If two numbers are in the ratio 2:6 and their LCM is 180. Find the numbers.

Discover how to find two numbers in the ratio 2:6 with an LCM of 180 using step-by-step methods, including prime factorization. Learn the process and concepts behind ratios and Least Common Multiple calculations, suitable for students in Grades 6-8.

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