: Two numbers are in the ratio 2 : 3 and their LCM is 180. What is the HCF of these numbers?

Step 1: Let's denote the two numbers as 2x and 3x, where x is a common factor.

Step 2: Since the LCM (Least Common Multiple) of two numbers is the product of the numbers divided by their HCF (Highest Common Factor), we can set up the following equation using the given LCM of 180:

2x * 3x / HCF = 180

Step 3: Simplifying the left side of the equation gives:

6x^2 / HCF = 180

Step 4: To find the HCF, we need to know the value of x. We can find this by realizing that the HCF of the two numbers (2x and 3x) is x itself. This is because 2 and 3 are prime numbers and do not have any common factors other than 1, so the HCF must be the common factor we introduced, x.

Step 5: Substituting HCF = x into the equation gives:

6x^2 / x = 180

Step 6: Simplifying this gives:

6x = 180

Step 7: Solving for x gives:

x = 180 / 6 = 30

Step 8: Therefore, the HCF of the two numbers is 30.