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Number Systems For CAT Exam Concepts, Shortcuts, Syllabus

In this article, you will learn about number systems for CAT tricks,  concepts and number system topics for CAT. By IIM Skills- online CAT coaching


Number Systems For CAT is a favorite section in the exam but it also well-received by the students. The topics are well-known for engaging conceptual puzzles that test the most suitable brains. For this topic, you are required to employ a mixture of theoretical mathematical principles and logical abilities. 



  • Number Systems For CAT concepts
  • Number system topics 
  • shortcuts techniques for number systems in CAT
  • best books for quantitative aptitude in CAT
  • Number Systems For CAT practice questions


Number Systems For CAT Introduction


This is a subject that starts at the pure basics of mathematics, which is, numbers. It explains to the students the different types of numbers so we can make application of these concepts to a variety of areas.


Here is the list of the Top CAT Coaching Institutes In Delhi 


CAT Number System Syllabus


The number system is such a large field that the accurate syllabus for it cannot be determined. Keeping in mind the number of system topics for CAT from which questions have appeared in the exam in recent years, the main number system topics for CAT include:

  • LCM and HCF
  • Prime and composite numbers
  • Properties of prime numbers
  • Theorems on prime numbers like Fermat’s theorem, Wilson’s theorem, remainder theorem, Euler’s theorem
  • Factorial of numbers
  • The number of zeroes in n!
  • Number of factors
  • Sum of the factors
  • Number of odd or even factors
  • Number of positive integral solutions
  • Divisibility rules
  • Divisibility properties
  • Cyclicity 


These were the topics that have appeared in the Number Systems For CAT section in recent years.


Number system for CAT tricks:


You should know and learn the basic purpose and concepts of numbers before we learn the number system for CAT tricks and tips. You also have to be mindful when you attempt a problem and identify the question from the number type. There is a notable variation between natural numbers, real numbers, and integers. 


Make sure to read and remember all prime numbers till 100.


Difficult topics such as the number of zeroes or the highest power, unit’s place digit, digital root, and Euler number have to be covered completely. 


A number system for cat tricks and an adequate number of questions in each of these topics need to be solved to get skilled in the field. Remember, whenever you do exercise questions for topics such as factorial, ensure that all the processes of solving that problem are clear to you.


  • Every number has the same digit at its unit’s place in its fifth power as it has at its first power, thus the standard process to find the digit at the unit’s place is to divide the power by 4, find the remainder power and find the unit’s digit in that number. This shortcut technique works because the digits at the unit’s place follow a cycle.


  • To find out the number of zeroes at the end of the factorial of any number, divide the number by 5, the quotient attained is again divided by 5 and repeated until the last quotient obtained is smaller than 5. The sum of all the quotients is the number of 5s, which then becomes the number of zeroes in the given number.


  • The digital root of any number is the sum of its digits, repeated until it becomes a single-digit number. For instance, the digital root of 87983 is 8 + 7 + 9 + 8 + 3 ⇒ 35 = 3 + 5 ⇒ 8.


  • Identify where the concept of Euler number can be used, and where the dividend and divisor are found to be co-prime, the remainder questions become very simple.


  • “The product of 3 consecutive natural numbers is perfectly divisible by 6.”


  • “The product of 3 consecutive natural numbers, the first of which is an even number is perfectly divisible by 24.”


  • “The sum of a two-digit number and a number formed by reversing its digits is perfectly divisible by 11.” Example, 27 + 72 = 99, is divisible by 11. Another fact is that the difference between those numbers will be perfectly divisible by 9. e.g. 99– 27 = 72, which is divisible by 9.


  • “∑n = n(n+1)/2, ∑n is the sum of first n natural numbers.”


  • “∑n2 = n(n+1)(n+2)/6, ∑n2 is the sum of first n perfect squares.”


  • “∑n3 = n2(n+1)2/4 = (∑n)2, ∑n3 is the sum of first n perfect cubes.”


  •  “xn + yn = (x + y) (xn-1 – xn-2.y + xn-3.y2 – … +yn-1) when n is odd. Therefore, if n is odd, xn + yis perfectly divisible by x + y.”


  • “xn – yn = (x + y) (xn-1 – xn-2.y + … yn-1) when n is even. Therefore, when n is even, xn – yn is divisible by x + y.”


  • “xn – yn = (x – y) (xn-1 + xn-2.y + …. + yn-1) for both odd and even n. Therefore, xn – yn is divisible by x – y.”


CAT Number System practice Questions:


Here are some practice questions for Number Systems For CAT to clear some basic concepts.


Example 1: N = (18n2 + 9n + 8)/n; where N is an integer. How many integral values can N have?




The expression can be broken as: 

⇒ 18n2/n + 9n/n + 8/n. 

⇒ This gives us: 18n + 9 + 8/n. 

Now we see that for all integral value of ‘n’, 18n + 9 will always return an integer. 

⇒ Hence, it depends upon 8/n 

⇒ n can have any integer number that is a factor of 8. 

⇒ The integers that satisfy this condition are ±1, ±2, ±4 and ±8

⇒ Thus, in total,n can have 8 possible values.


Example 2: N = 960. What is the number of factors of N?



We see that N is a composite number

Let D be a composite number in form D = ap × bq × cr, where a, b, c are prime numbers, 

⇒ total divisors of D, given by n is = (p+1)(q+1)(r +1). 

⇒ similarly, after dividing 960 into prime factors: 26 × 31 × 51, we can determine the total number of factors as (6+1) X (1+1) X (1+1) = 28.


Example 3: Find the unit’s place digit of the following: (123)34 × (876)456 × (45)86.



We see that since there is no 5 in the unit’s place

Whenever an even unit digit and a 5 at the unit digit are present, they will always give a 0 at the unit digit, no matter if any other number is present or not. 

Hence, this approach would be best.

In the second number, the unit’s digit will always be 6.

Similarly in the third number, the unit’s digit will always be 5

So according to the principle discussed

6 X 5 = 30

Hence the unit’s digit is 0.


Example 4: Find the number of “zeroes at the end of the product of the first 100 natural numbers”?



In these sorts of questions, you need to find the highest power of 5, which can divide the product of the first 100 natural numbers. 

We know, any multiple of 5 multiplied by any even number, results in a zero at the unit’s place. 

Divide 100 by 5 and which gives 20 as quotient. 

Then divide this 20 (the quotient) by 5 and the new quotient comes as 4, 

4 cannot be further divided by 5. 

The sum of all these quotients gives us the highest power of 5, which can divide that number. 

The sum comes as 24 which is the answer to the question.


Example 5: Which letter should replace the $ in the number 2347$98, so that it becomes a multiple of 9?



To solve this question we need to use the principle of divisibility by 9.

We know that “if the sum of all the digits is divisible by 9, then the number is divisible by 9.”

Now sum of the given digits is 2 + 3 + 4 + 7 + 9 + 8 = 33 + $. 

We need the next multiple of 9 after 33 

That is 36. 

This means the value of $ is 3.


Example 6: At a party, there are 20 people. If each of these people shakes hand with every other person, how many total handshakes take place?



Out of 20 persons, the first person shakes hands with 19 other people. 

The second person shakes hands with 18 other people (because the second person and first person handshake have already happened). 

The third person will similarly shake hands with 17 other people, 

and so on. 

The second last person shakes hand with only one person. 

And last shakes hand with none (as already shaken hand with everyone). 

To determine the total number of handshakes, we just have to add all the natural numbers from 1 to 19 i.e. ∑ 19. 

∑19 = 19 x 20/2 = 190 total handshakes. (see the rule above)


Question 7: The sum of all factors of X is 124. What is the value of X?


  1. X lies between 40 and 50
  2. X lies between 50 and 60
  3. X lies between 60 and 80
  4. More than one X exists



Method of solving this question from Number Theory – Factors: In some situations, going back would be really difficult.

“Any number of the form paqbrc will have (a + 1) (b + 1)(c + 1) factors, where p, q, r are prime.”


“For any number N of the form paqbrc, the sum of the factors will be (1 + p1 + p2 + p3 + …+ pa) (1 + q1 + q2 + q3 + …+ qb) (1 + r1 + r2 + r3 + …+ rc).”


Sum of factors of number X is 124. 

124 can be represented as 22 * 31. 

or 4 * 31, or 2 * 62 or 1 * 124.

2 cannot be rewritten as (1 + p1 + p2 + p3 + …+ pa) for any value of p.

But 4 can be represented as (1 + 3)

So, we need to see if 31 can be written in that form.

The interesting point to be seen here is that 31 can be represented in two different ways.

31 = (1 + 21 + 22 + 23 + 24)

31 = ( 1 + 5 + 52)

Or, the number X can be represented as 3 * 24 or 3 * 52. Or X can be 48 or 75.


Therefore the answer is D.


Question 8: How many factors of the number 1080 are perfect squares?


  1. 4
  2. 6
  3. 8
  4. 5


We know the method to find the number of factors of any number. 

But How do we check if they are perfect squares?

1080 = 23 * 33 * 5. 

For any number that is a perfect square, all the powers/exponents of the prime numbers need to be even. 

So, if the factor can be of the form 2a * 3b * 5c.

The possible values ‘a’ are 0 and 2, 

Possible values of b are 0 and 2, 

and possible values of c are 0.

So in total, there are 4 possibilities. 1, 4, 9, and 36.


Hence A is the correct answer.


Question 9: How many factors of 2 * 53 * 74 are odd numbers?


  1. 100
  2. 99
  3. 20
  4. 24


We know how to find all number of factors

Any factor of this number should be of the form 2a * 5b * 7c.

For the factor to be an odd number,

a must be 0.

The possible value of b =  0, 1, 2, 3.

The possible value of c =  0, 1, 2, 3, 4.

Total count of odd factors = 4 * 5 = 20.


Hence C is the correct answer.


best books for quantitative aptitude for CAT


How to Prepare for Quantitative Aptitude for the CAT by Arun Sharma and Meenakshi Upadhyay


A short introduction to the authors first. 


Arun Sharma is an alumnus of the Indian Institute of Management, Bangalore. He has been guiding corporates and CAT applicants for more than 2 decades and has personally trained over a thousand students who later got admitted into the IIMs and other top-ranked Business Schools. 


He also holds the record of cracking the CAT exam 16 consecutive years, scoring a percentile of 99.99 in CAT 2008.


Meenakshi Upadhyay is an alumnus of IIM Bangalore and is actively involved in training for the CAT, CSAT and many other management tests for more than 15 years.


The book highlights complete theory according to the latest syllabus and pattern of CAT. The subjects are broken down into small pieces for the benefit of the student. The book also has lots of shortcut methods that may be convenient during mock and the main exam.


A section called ‘taste of exams’ at the end of each topic has questions from that topic that have appeared in previous years papers of exams like CAT, XAT, IIFT, etc. in the last 20-25 years.


The most beneficial fact about this book is that the theory is descriptive and explained with patience. The author was aware of the requirements of a student who studies by himself/herself and has taken responsibility for as many of them without being physically present with the candidate. 

This book is one of the best books for the preparation of CAT. If you use just this book, you will not need any other book for the quantitative aptitude section of CAT. This book gives full preparation of quant in CAT.


The book is divided into different sections and has these features:

  • Elaborate theory
  • Thousands of practice questions
  • Practic questions of different difficult level
  • Mock tests
  • Previous years questions CAT


Quantitative Aptitude for All Competitive Examinations by Abhijit Guha


This is a good book for the preparation of not just CAT but many other competitive exams that have a quant section. These exams include SBI PO and IBPS too. 


Since exams like SBI PO are raising their levels in recent years, the difficulty level of SBI PO is coming closer and closer to CAT.


Nonetheless, this is a useful book for the preparation of quantitative aptitude. But there are a few problems that the students who bought this book had, most importantly the absence of solutions in many easy questions. It was considered by the author that some easy questions did not need detailed solutions, but this was not good for the candidates who need to start from scratch.


Some students found it too easy compared to the level of CAT, however solving easy questions is also important since it makes you comfortable with the basics and improves speed and skill that is important for the tests.


The book has a huge collection of practice questions. This will help also in the placement season if you happen to be in the final year of your engineering since many applicants in CAT are engineers. 


Quantitative Aptitude Quantum CAT by Sarvesh Verma


This book is very suitable for those applicants who are preparing for management school entrance exams. The book is divided into topic-wise sections. First, basic ideas and concepts are explained and then there are practice questions.

The solutions are properly described with principles and theories. The book has all the varieties of questions from topics that have been asked in CAT and other exams yet. 

The book’s print and page quality may be problematic but its content is absolutely good. This book by Sarvesh Verma is probably an answer to all the concepts for the quant section in CAT.


Quantum CAT by Sarvesh Verma has chosen the method of instruction through examples. The book includes more than 300 concepts and more than 4000 quant questions with descriptive solutions. 

The book explains many shortcut techniques required for saving time during the CAT.


The book includes these sections:

CAT solved papers

XAT solved papers

IIFT solved papers

SNAP solved papers

Fundamentals, Averages, Allegations, Ratio, Proportion & Variation, Percentages, Profit, Loss & Discount, CI/SI/Installments, Mensuration, Logarithm, Functions and Graph, Sequence and Series and Progressions, Elements of Algebra, Theory of Equations, Set Theory, Time and Work, Time, Speed and Distance, Permutations & Combinations, Probability, Trigonometry, Geometry, and Coordinate geometry.


Why is quantitative ability in CAT?


You must be thinking why is there a section of Number Systems For CAT. In fact, why is there a quantitative section in CAT at all? Let us see why.


Quantitative ability is a very crucial aspect of a person’s personality. By quantitative ability, we usually refer to the skill that basically decides our analytical and problem-solving abilities. It relates to your individual abilities as a human – and how properly you are able to apply them to particular questions in CAT.


This is why it’s one of the most significant elements of competitive exams and job interviews as well. You can find aptitude questions in virtually every exam paper. Why? The purpose is to assess your problem-solving and decision-making abilities – which are important for your academic/professional achievement in the long term.


In short, the quantitative ability is what defines the analytical and applicational strength of an individual, as it is very much necessary for an individual to think critically in complicated conditions. This is more obvious nowadays than ever. While you can recognize quantitative ability as being fundamental to every recruitment team’s evaluation of you. This is because quantitative aptitude is what foretells how you will perform in real work.


It actually displays the following skills:


  • Your foresight and observation potential.
  • Your Memory, Attention, and Calculations.
  • Your common sense and situational awareness.


That’s why quantitative aptitude in CAT is so important to your performance as a student: they are a record of your competence—a proof of your quantitative aptitude as a candidate.


Having said that, there is far more to the individualistic nature of the individual than we care to consider. You, as a person, are much more than your IQ – which somewhat connects to your performance on a quantitative aptitude test. 


Equally important is your “EQ” (ie.  emotional quotient), and your social skills. We call them soft skills. These are constituents beyond the range of what composes quantitative aptitude– but they equally as, if not more, significant as your quantitative aptitude.


An individual with a high quantitative aptitude score in CAT but a poor capability to express his ideas effectively and/or work in a team is objectively far less likely to achieve success than any individual with a moderate quantitative aptitude score but excellent social skills and a high EQ. So here, more than anywhere else, a balance is important.


Nevertheless, most academic/professional organizations do value a good quantitative aptitude test score profoundly – so you mustn’t totally disregard it, even if you think it is not the most important feature of what makes you. 


For more helpful articles like the number system for CAT, visit the IIM Skills CAT blog.


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Gaurav is a Content Writer at IIM Skills. He has a B.Tech. degree but then he switched to the creative side by doing his master's in advertising and public relations. Gaurav is also a part-time blogger and graphic designer currently living in Mumbai

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