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

Step 1: Let's assume the two numbers are 2x and 3x.

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

LCM = (2x * 3x) / HCF

Step 3: We know that the LCM is 180, so we can substitute that into the equation:

180 = (2x * 3x) / HCF

Step 4: Simplifying the right side of the equation gives us:

180 = 6x^2 / HCF

Step 5: To find the HCF, we need to know the value of x. However, we don't have enough information to find x.

Therefore, we cannot find the HCF of the two numbers with the information given.