### Numbers

Number System is an important topic that is covered all the time in MBA exams. The reason why Numbers System is being included in the QA part of these exams is that MBA aspirants are supposed to have a very good calculation mindset when it comes to working in a corporate job

Below, the details about the Number System under QA for different competitive MBA exams are given:-

#### CAT :

 Number System Year No of Questions Good attempt Difficulty 2019 Slot 1 1 – Difficult Slot 2 3 1 Difficult 2018 Slot 1 4 4 Easy Slot 2 4 2 to 3 Moderate 2017 Slot 1 4 3 Easy Slot 2 4 3 Easy

XAT :

 Number System Year No of Questions Good attempt Difficulty 2020 2 1 Moderate 2018 2 1 Easy – Moderate

SNAP :

 Number System Year No of Questions Good attempt Difficulty 2019 3 2 Easy-Moderate 2017 5 2 to 3 Moderate

### List of Concepts in that Chapter

The list of concepts that are covered in the Number System chapter is as follows: –

• Algebraic Formulas
• Law of indices
• Binomial Theorem
• Finding out recurring/non-recurring decimal
• Converting recurring/non-recurring decimal to fractions and vice versa – For e.g: – “Convert 0.125125125… to the fractional form”
• Converting mixed recurring decimals to fractions – For e.g: – “Express 0.18888… as a fraction”
• Finding out the last digit/unit’s digit or last two digits– For e.g: – “What is the last digit of A multiplied by B?”
• Finding out the remainder- For e.g: – “Find out the remainder when 7^25 is divided by 6”
• Interchanging of digits – For e.g: – “The digits of a two-digit number are in the ratio of 2 : 3 and the number obtained by interchanging the digits is bigger than the original number by 27. What is the original number?
• Checking the divisibility rules- For 2,3,4,5,6,7,8,9,10,11,13,17,19,23
• Finding out the number of factors- For e.g: – “How many factors of (2^5) * (3^6) * (5^2) are perfect squares?”
• Finding out the number of zeroes – For e.g: – ”What is the number of trailing zeroes in 1123!?”
• Finding out the highest power- For e.g: – “Find the largest power of 3 contained in 95!”
• Base Conversion – From decimal to other bases or vice versa
• Least common multiple (LCM) and Highest Common Factor (HCF) – For e.g – “Find the least number, which is exactly divisible by x,y,z”
• Also, there will be questions from whole numbers, real numbers, natural numbers, composite numbers, rational/irrational numbers, even/odd numbers, and prime numbers

### Some Questions from Previous Papers

Some of the important CAT questions on number system that appeared in the previous papers are: –

1. How many factors of (2^4) *(3^5) *(10^4) are perfect squares which are greater than 1? (CAT 2019 – SLOT 2)

24 x 35 x 104 = 2^4×3^5× (2^4×5^4 ) = 2^8×3^5×5^4
Perfect square factors of the number will be of the form 2^a×3^b×5^c
Where a can be 0, 2, 4, 6 or 8. b can be 0, 2 or 4 and c can be 0, 2 or 4
so total number of square factors = 5*3*3 = 45
But when a=b=c =0 factor will be 1.
Thus, perfect squares which are greater than 1= 45 -1 = 44

2. What is the largest positive integer n such that ( (n^2) + 7n +12)/( (n^2)-n-12) is also a positive integer? (CAT 2019 – SLOT 2)

Given,
(n^2+ 7n+12)/(n^2-n-12 )=(n^2+ 3n+4n+12)/(n^2-4n +3n-12)
=((n+3)(n+4))/(n-4)(n+3)
=(n+4)/(n-4) is an integer
From the option we can see largest possible value of n =12

3.Let a,b,x,y be real numbers such that a^2 + b^2 = 25, x^2 + y^2=169, and ax+by=65. If k=ay-bx, then k=? (CAT 2019 – SLOT 2)

As a,b, x , y are real and as we know 3^2 + 4^2 = 25
Or 5^2 + 0= 25 also 13^2 + 0= 169 and 5^2 + 12^2 = 169
ax+ by = 65 is possible only when (a,b) = (0,5) and (x,y)= (0,13)
Thus k = 0*13 – 0*5 =0

4. The number of integers x such that 0.25 < 2^x < 200, and 2^x +2 is perfectly divisible by either 3 or 4, is? (CAT 2018 – SLOT 1)

As given 0.25 < 2x < 200
Or ¼ < 2^x < 200
So x = { -2, -1, 0,1,2,3,4,5,6,7}
Now 2^x +2 will be perfectly divisible by 3 if x is even non-negative integer and will be divisible by 4 if x = 1
So number of possible solution = 5 { x = 0, 1, 2, 4 ,6}

5. How many numbers with two or more digits can be formed with the digits 1,2,3,4,5,6,7,8,9, so that in every such number, each digit is used at most once and the digits appear in the ascending order? (CAT 2018 – SLOT 1)

As the digits appear in ascending order in the numbers, number of ways of forming a n-digit number using the 9 digits
= 9Cn
Number of possible two-digit numbers which can be formed = 9C2+9C3+9C4+9C5+9C6+9C7+9C8+9C9
=(2^9)−(9C1+9C1)=(2^9)−(9C1+9C1)
=512−(1+9)=502

### How to deal with Number preparation

Level 1

• Students are supposed to have basic knowledge of the properties of numbers
• They should have a fast calculation mindset
• Students need to have the skill to identify which type of number system is being used and find solutions
• Students need to avoid their calculators and practice calculations using hand and develop speed

Level 2

• Topics such as highest power, last digit, number of factors, interchanging of digits are important and students need to be thorough with these chapters
• Questions from remainders can be a little twisted, so advising students to practice this chapter carefully. Also, spending too much time is harmful, so try to answer questions that are more straightforward and keep twisted questions to the end

Level 3

Students need to look into problems that use multiple concepts to find a solution. Try to answer questions from the advanced level of previous CAT papers, mock tests, and materials. And, try to attempt these questions in a time-based manner. Taking too much time for advanced questions will be harmful to crack the exam. Finally, students need to practice as many questions as they can to get a hold of the concepts and find solutions fast. Practice will always help the student to crack the exam