MBAP logo

How many pairs (a, b) of positive integers - Quantitative Aptitude - Equation

CAT 2020 – The CAT QA section has grown increasingly difficult since 2015. In order to tackle the tougher CAT Level QA questions for the CAT Exam, it is important to understand the basics of CAT. To obtain a great CAT score, make use of MBAP Free Study material with detailed solutions and video explanations. Check out MBAP free Mock test to take these questions in a test format for free.

The question below is from previous year CAT question from CAT 2020 exam comes from CAT Equation : How many pairs (a, b) of positive integers

Find out by answering this question which tests an aspirant’s Quantitative Ability Skills.

You may also find remaining question solution of CAT 2020, Slot 3 by searching the question in the search bar.

CAT 2020 - Slot 3 - Quantitative Aptitude - Equation - Question 26 - How many pairs (a, b) of positive integers

Q. 26: How many pairs (a,b) of positive integers are there such that a ≤ b and ab = ?

1. 2019
2. 2018
3. 2020
4. 2017

26. B.

Number of factors of  = 2×2017 + 1= 4035

Number of pairs of a,b such that a×b =  is equal to half the Number of factors of 

Number of pairs of a,b such that a×b = 

= 4035/2

You observe that 4035/2 is not an integer, because 22×2017 is a perfect square and one of the pairs of (a,b) is (2017 , 2017).

So, the number of such pairs is the highest integer roundoff of 4035/2 = 2018.

Among all these 2018 pairs, one of the integer is less than or equal to the other. (Equal in the case of (2017,2017))

We assume that a is the least one of the two…

Hence there are 2018 pairs that satify this condition.

 

Alternate Method:

a × b = 

a × b = 

 

a and b are of the form 2x and 2y respectfully…

Since a ≤ b;

x ≤ y

Also, a × b = 

2x + 2y = 

x + y = 2×2017

 

We have 2 conditions to deal with…

x ≤ y and x + y = 2×2017

Let’s start with x = y:

Here, x = y = 2017

 

From here on, we decrement x and increment y to maintain the conditions x ≤ y and x + y = 2×2017

We can keep doing this until x = 0, because if x is negative, a which is 2x will not remain an integer.

 

Hence x can range from 2017 to 0; and thereby x can take 2018 values.

Therefore, there can be 2018 pairs of (a,b) that sitisfy a ≤ b and ab = 

 

Past Year Question Paper & Solutions

Counselling Session
By IIM Mentor

Reach Us

primary Menu

Inspiring Education… Assuring Success!!
Ⓒ 2020 – All Rights Are Reserved

Free Content !!!
Signup now to get the All recent CAT Exam Paper with Solution... Subscribe below with your Email ID
Free Content !!!
Signup now to get the All recent CAT Exam Paper with Solution... Subscribe below with your Email ID