Concept: The question tests on HCF/LCM as a DS question

Solution:

This is a very simple question on the concepts of HCF and LCM. From the question data, we know that x and y are positive integers. We need to find the value of xy.

From statement I alone, HCF(x,y) = 10. This is insufficient since x and y can take a host of values such that their HCF is 10.

For example, x = 10 and y = 10 in which case their HCF is 10 and their product is 100.
x = 10 and y = 20 in which case their HCF is still 10 but their product is 200.

Statement I alone is insufficient to give us a unique value for xy. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, LCM(x,y) = 180. This is insufficient since x and y can take a host of values such that their LCM is 180.

For example, x = 36 and y = 5 in which case their LCM is 180 and their product is 180.
x = 90 and y = 20 in which case their LCM is 180 but their product is 1800.

A common mistake that some test takers make here is to remember the data from the first statement and therefore conclude that the first case is not possible. This is not right, note that you are trying to solve the question using the second statement alone.

Statement II alone is insufficient. Answer option B can be eliminated. Possible answer options are C or E.

Combining statements I and II, we have the following:
From statement I, HCF (x,y) = 10; from statement II alone, LCM (x,y) = 180.

Combining these two pieces of information should be done using a property rather than plugging in values. Remember, you are not trying to find the values of x and y individually; you are trying to calculate xy.

For any two numbers, Product of numbers = Product of their LCM and HCF.

Therefore, xy = 180 * 10 = 1800.
The combination of statements is sufficient to find a unique value of xy. Answer option E can be eliminated.

 The correct answer option is C.