CAT 2020 Question Paper | Quants Slot 3

CAT Questions

The best questions to practice for CAT Exam are the actual CAT Question Papers. 2IIM offers you exactly that, in a student friendly format to take value from this. In CAT 2020 we saw some beautiful questions that laid emphasis on Learning ideas from basics and being able to comprehend more than remembering gazillion formulae and shortcuts. Original CAT Question paper is the best place to start off your CAT Preparation. Here's one interesting question from CAT 2020 Question Paper with a detailed video and text solutions. To check out about 1000 CAT Level questions with detailed video solutions for free, go here: CAT Question Bank

Question 26 : How many pairs (a,b) of positive integers are there such that a ≤ b and ab = 4²⁰¹⁷?

---

---

Video Explanation

---

---

Explanatory Answer

4²⁰¹⁷ = 2^2×2017
Number of factors of 2^2×2017 = 2×2017 + 1= 4035
Number of pairs of a,b such that a×b = 2^2×2017 is equal to half the Number of factors of 2^2×2017
Number of pairs of a,b such that a×b = 2^2×2017
= \\frac{4035}{2})
You observe that \\frac{4035}{2}) is not an integer, because 2^2×2017 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 \\frac{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 = 4²⁰¹⁷
a × b = 2^2×2017

a and b are of the form 2^x and 2^y respectfully...
Since a ≤ b;
2^x ≤ 2^y
x ≤ y
Also, a × b = 2^2×2017
2^x + 2^y = 2^2×2017
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 2^x will not remain an integer.

x	y
2017	2017
2016	2018
2015	2019
⁝	⁝
⁝	⁝
⁝	⁝
2	4033
1	4034
0	4035

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 = 4²⁰¹⁷

---

The question is "How many pairs (a,b) of positive integers are there such that a ≤ b and ab = 4²⁰¹⁷?"

Hence, the answer is, "2018"

Verbal Ability for CAT | CAT Reading Comprehension

Verbal Ability for CAT| CAT Para Jumble

Verbal Ability for CAT | CAT Sentence Elimination

Verbal Ability for CAT | CAT Paragraph Completion

Verbal Ability for CAT | CAT Critical Reasoning

Verbal Ability for CAT | CAT Word Usage

Verbal Ability for CAT| CAT Para Summary

Verbal Ability for CAT | CAT Text Completion

Verbal Ability for CAT | CAT Sentence Correction