Number System for CAT: Concepts, Shortcut methods, Practice Questions and Previous Year Questions

CAT QA Number System is one of the most important topics in the Quantitative Aptitude section. It is one of the most challenging topics to prepare and attempt. The questions cover a wide range of topics like remainder theorem, unit digit, divisibility, prime numbers, number of zeros and rational numbers.

• CAT QA section consisted of 34 questions before 2022. However in 2020, the pattern changed and only 26 questions were asked, out of which 3-4 questions were from the Number System. The total number of questions in CAT 2021 were further reduced to 66 (22 from QA).
• Candidates should make sure that they prepare this section well as it accounts for about 10% of the total questions in CAT Quantitative Aptitude section.
• For CAT 2022, Candidates must refer to books like Quantum CAT by S K Verma, Quantitative Aptitude by R S Aggarwal and Quantitative Aptitude for CAT by N K Sinha.
• As per toppers the best way to score maximum marks in this section is to practice at least 30-40 questions per day.

Important Number System Concepts for CAT Quant Section

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CAT QA Number System : Important Concepts and Formulae

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The syllabus for the number system cannot be clearly defined, however as per the analysis of trends in CAT Question papers it can be observed that the following topics are the most important as they are repeatedly asked with varying levels of difficulty.

Candidates should not miss out on these topics and must prepare them thoroughly.

The questions basically come from these topics and can range from the basic level to more application based.

CAT QA Number System: HCF and LCM

HCF and LCM are two topics that need to be mastered by every candidate to score a high percentile in CAT as this is important for questions of Number System and builds a base for other sections as well, especially arithmetic and algebra.

Important points to remember:

1. HCF * LCM = Product of numbers(a*b)
2. There are 2 methods for finding LCM - Factor method and Division method.
3. The HCF is the factor of the difference of the given numbers.

For e.g. HCF of 30 and 42

Difference = 12(42-30)

Factors of 12 in descending order - 12, 6, 4, 3, 2, 1

Divide both 30 and 42 by the factors in the above order to check which number divides both completely, that number is your HCF. 

CAT QA Number System: Concept of Unit Digit

The one’s digit of any number is called the unit digit of that number. Finding the unit digit of a given expression(product, power) is not so difficult. One should remember the points mentioned below for clarity:

1. Use only the unit digits of each number in the given expression to find the unit digit of the complete expression.

For e.g. to find the unit digit of 676 * 453 * 39,

just multiply the unit digits of each number

i.e. 6 * 3 * 9 = 162. So, 2 is the answer.

1. Concept of Cyclicity Cyclicity of numbers refers to the period after which the number repeats itself in a power. Find below the table of cyclicity:

CAT QA Number System : Divisibility

Candidates must read the important points mentioned below for understanding the concept:

1. Dividend = (Divisor * Quotient) + Remainder
2. If X and Y are two numbers and are completely divisible by divisor D , then the given rules hold true:

• X+Y and X-Y are also divisible by D
• Product of X or Y with any other integer is also divisible by D

1. Rules of Divisibility:

CAT QA Number System : Concept of Remainders

For CAT, it is extremely important to prepare the concept of Remainders. The below results must be kept in mind while preparing the same:

1. The respective remainders obtained by dividing numbers x1, x2, x3……. Individually by divisor D be r1, r2, r3,…... , then :

• Remainder for x1 + x2 + x3 +........ can be directly obtained by dividing r1 + r2 + r3 +..... by D
• Remainder for x1 * x2 * x3 *..... can be obtained by dividing r1 * r2 * r3 *..... by D

1. {(a+1)ⁿ / a} always gives remainder 1.

CAT QA Number System : Number of Zeros

The number of zeros at the end of the product of any numbers is determined by the number of combinations of ‘2 * 5’ provided the below conditions are satisfied:

• 2n * 5m gives n number of zeros if m>n
• 2n * 5m gives m number of zeros if n>m

CAT QA Number System : Integers

Every candidate should know the basic rules and properties of integers for improving their attempts in the CAT exam.

• (-a) + (-b) = -(a+b)
• -a * b = -ab
• -a * -b = ab
• (-a) + b = b-a
• (-a) - (-b) = (-a) + b = b-a
• Multiplicative identity of all integers is 1
• Additive identity of all integers is 0.

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CAT QA Number System : Tips and Tricks

These tips and strategies are of utmost importance to be able to attempt all questions on the CAT Exam day.

1. Learn the square of numbers till 35 and cube of numbers till 20.
2. Ensure that you memorize all prime numbers upto 100.
3. Make sure that you are aware of the basics of all the types of numbers and the differences between them. These include Whole numbers, Natural numbers, Integers, Real numbers, Irrational numbers, Rational numbers, Complex numbers.
4. Practice more and more questions everyday making sure that you attempt at least 35-40 questions from topics like HCF and LCM, Factorials, highest power, remainder, unit’s digit etc.
5. Try to make notes of topics you find difficult in a comprehensive manner so that you can revise these topics easily whenever you need.
6. Make sure you follow only 1 book when you start preparing for the CAT Exam. However, once you are done with this book you can use other books to supplement your previous knowledge and practice.
7. Ensure that you practice questions on the above topics from CAT Previous Year papers.

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CAT QA Number System : Shortcut Methods

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Make sure you follow these given shortcut methods for various topics to excel in the exam:

• Sum of first n natural numbers = [n(n+1)] / 2
• Sum of squares of first n natural numbers = [n(n+1)(2n+1)] / 6
• Sum of cubes of first n natural numbers = [n(n+1)]² / 4
• Sum of first n even natural numbers = n(n+1)
• Sum of first n odd natural numbers = n²
• Sum of squares of first n even natural numbers = [2n(n+1)(2n+1)] / 3
• Sum of squares of first n odd natural numbers = [2n(2n+1)(2n-1)] / 3
• The calculation of digital sum is very easy to understand. It simply refers to the process of repeated addition of the digits of the number until a single digit is left. For example, the digital sum of 34678 will be 3 + 4+ 6 + 7 + 8 = 28 = 2 + 8 = 10 = 1 + 0 = 1. Hence, 1 is the digital root of 34678.
• When (x-1)ⁿ is divided by x, then the remainder is:

1, if n is an even natural number

x-1, if n is an odd natural number.

• The shortcut trick for finding the unit digit of a given number of the form Xⁿ is to divide power ‘n’ by the cyclicity of the number X and find the unit digit for the remaining power of the number X.

For e.g. for finding the unit digit of 2³⁵

Divide 35 by 4(cyclic power of 2)

The remaining number will be 3

So, the unit digit will 2³ = 8

• When the digits of a 3- digit number are reversed; then the difference between the given number and the reversed number is always divisible by 99.