The question is about HCF and LCM. We need to find all the possible pairs of (x,y), given the product of them and their HCF. Dealing with HCF and LCM of numbers is a vital component in CAT Number Theory: HCF and LCM. A range of CAT questions can be asked based on this simple concept. In CAT Exam, one can generally expect to get 1~2 questions from CAT Number Systems: HCF and LCM. In multiple places, extension of HCF and LCM concepts are tested by CAT Exam and one needs to understand HCF and LCM to be able to answer the same. CAT Number theory is an important topic with lots of weightage in the CAT Exam.

Question 1: How many pairs of integers (x, y) exist such that the product of x, y and HCF (x, y) = 1080?

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9

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We need to find ordered pairs (x, y) such that xy * HCF(x, y) = 1080.

Let x = ha and y = hb where h = HCF(x, y) => HCF(a, b) = 1.

So h^{3}(ab) = 1080 = (2^{3})(3^{3})(5).

We need to write 1080 as a product of a perfect cube and another number.

Four cases:

1. h = 1, ab = 1080 and b are co-prime. We gave 4 pairs of 8 ordered pairs (1, 1080), (8, 135), (27, 40) and (5, 216). (Essentially we are finding co-prime a,b such that a*b = 1080).

2. h = 2, We need to find number of ways of writing (3^{3}) * (5) as a product of two co-prime numbers. This can be done in two ways - 1 and (3^{3}) * (5) , (3^{3}) and (5)

number of pairs = 2, number of ordered pairs = 4

3. h = 3, number of pairs = 2, number of ordered pairs = 4

4. h = 6, number of pairs = 1, number of ordered pairs = 2

Hence total pairs of (x, y) = 9, total number of ordered pairs = 18.

The pairs are (1, 1080), (8, 135), (27, 40), (5, 216), (2, 270), (10, 54), (3, 120), (24, 15) and (6, 30).

The question is **"How many pairs of integers (x, y) exist such that the product of x, y and HCF (x, y) = 1080?"**

Choice C is the correct answer.

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