CAT Quantitative Aptitude Questions | CAT Number Systems - HCF and LCM questions

The question is about HCF and LCM. We need to find out the possible pairs of (x,y), which satisfies the given conditions about the sum of their HCF and LCM. 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 5: How many pairs of positive integers x, y exist such that HCF (x, y) + LCM (x, y) = 91?

1. 10
2. 8
3. 6
4. 7

Explanatory Answer

Method of solving this CAT Question from Number Theory - HCF and LCM: Given a property of HCF and LCM of two numbers, how do we get to the underlying numbers.

Let us x = h * a; y = h * b
a and b are co-prime. So, LCM of (x, y) = h * a * b

So, in essence h + h * a * b = 91. Or h(ab + 1) = 91
Now, 91 can be written as 1 * 91 or 7 * 13
Or, we can have HCF as 1, LCM as 90 -
There are 4 pairs of numbers like this (2, 45), (9, 10), (1, 90) and (5, 18)

We can have HCF as 7, ab + 1 = 13 => ab = 12 => 1 * 12 or 4 * 3

Or, the pairs of numbers are (7, 84) or (21, 28)

The third option is when HCF = 13, ab + 1 = 7 => ab = 6
Or (a, b) can be either (1, 6) or (2, 3)
The pairs possible are (13, 78) and (26, 39)
There are totally 8 options possible - (2, 45), (9, 10), (1, 90), (5, 18), (7, 84), (21, 28), (13, 78) and (26, 39).
8 Pairs.

The question is "How many pairs of positive integers x, y exist such that HCF (x, y) + LCM (x, y) = 91?"

Hence the answer is "8 pairs"

Choice B is the correct answer.