CAT Number System is a vast topic that includes important sub-topics like unit's digit, reminders, digital root, divisibility rules, integers, and number of zeroes etc. Questions are also asked on real numbers, prime numbers, natural numbers, and whole numbers. Check CAT QA Syllabus

CAT Number System topic is known for judging conceptual problems that test the brain of the candidates. One must require a combination of pure mathematical knowledge and logic skills to be strong in this topic. In fact, this chapter begins with the absolute basics of mathematics, i.e, numbers. It explains the different types of numbers and then extends the application of this knowledge to a variety of domains. Check CAT QA Question Papers

Read the article to know more about tips to prepare for the number system for CAT 2021, and also check important formulas along with solved sample questions.

Basics of CAT Number System

Numbers are divided into two types:

1. Real Numbers
2. Imaginary Numbers

Real Numbers are further divided into:

1. Rational Number - have terminating or recurring decimal
2. Irrational Number - have non-terminating or non-recurring decimal

Further, the Rational Numbers are divided into:

1. Fractions
2. Integers

Integers are categorised into: 

1. Negative Integer 
2. Neither Negative nor Positive
3. Positive 

Other important types of Numbers to be Keep in Mind are:

1. Whole Numbers - 0,1,2,3,4 etc
2. Prime Numbers - which have only 2 factors i.e. 2,3,5,7 etc
3. Composite Numbers - which have more than 2 factors
4. 1 is neither a prime number or a composite number

Quick Links:

• CAT Quants Exam Pattern
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Important Topics from CAT QA Number System

The topics from Number System from which questions have appeared in CAT exam in recent years, include:

• LCM and HCF
• Prime and composite numbers
• Properties of prime numbers
• Theorems on prime numbers like 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

How to prepare for CAT Number System?

Number Systems is a vast topic and around 1-3 questions are asked in CAT exam. There are various topics under Number Systems. 

• Topics like, the number of zeroes or the highest power, unit’s place digit, digital root, and Euler number have to be covered completely.
• Number theory is a topic in which one needs to understand the basics: Factors, Multiples, etc and there are few solving techniques, tips, and shortcuts required which are discussed below. 
• But the exact way one can score good marks on this topic is by practicing. One must practice knowing all the rules so, practice enough questions.

CAT Preparation Tips for Prime and Composite Numbers

• Prime numbers are those which have only two factors, 1 and the number itself.
• Composite numbers have more than 2 factors. Examples are 4, 6, 8, 9.
• 0 and 1 are neither composite nor prime. There are 25 prime numbers less than 100.

Properties of Prime numbers

• List all prime factors less than or equal to √n to check if n is a prime number, l. 
• If none of the prime factors can divide n then n is a prime number.
• For any integer a and prime number p, ap−a is always divisible by p
• All prime numbers greater than 2 and 3 can be written as 6k+1 or 6k-1
• If a and b are co-prime then a(b-1) mod b = 1.
• Topics like, the number of zeroes or the highest power, unit’s place digit, digital root, and Euler number have to be covered completely.
• Number theory is a topic in which one needs to understand the basics: Factors, Multiples, etc and there are few solving techniques, tips, and shortcuts required which are discussed below. 
• But the exact way one can score good marks on this topic is by practicing. One must practice knowing all the rules so, practice enough questions.

Important Theorems on Prime numbers

• Fermat’s Theorem - Remainder of a^(p-1) when divided by p is 1, where p is a prime
• Wilson’s Theorem - Remainder when (p-1)! is divided by p is (p-1) where p is a prime
• Remainder Theorem - If a, b, c are the prime factors of N such that N= ap * bq * cr Then the number of numbers less than N and co-prime to N is
  • ϕ(N)= N (1-1/a) (1 – 1/b) (1 – 1/c).
  • This function is known as Euler’s totient function.

• Euler’s theorem - If M and N are co-prime to each other then the remainder when Mϕ(N) is divided by N is 1.

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Divisibility Rules to Check the Prime Numbers

Divisibility properties for CAT Number System

• For composite divisors, check if the number is divisible by the factors individually. So, to check if a number is divisible by 6 it must be divisible by 2 and 3.
• The equation an−bn is always divisible by a-b. If n is even it is divisible by a+b. If n is odd it is not divisible by a+b.
• The equation an+bn is never divisible by a-b. If n is odd it is divisible by a+b. If n is even it is not divisible by a+b.
• A decimal number is divisible by b-1 only if the sum of the digits of the number when written in base b are divisible by b-1.

Rules of Cyclicity in CAT Number System

To find the last digit of an, find the cyclicity of a. For Ex. if a=2, we see that

▸21=2 ▸22=4 ▸23=8 ▸24=16 ▸25=32

Hence, the last digit of 2 repeats after every 4th power. Hence cyclicity of 2 = 4. Hence if we have to find the last digit of an, The steps are:

• Find the cyclicity of a, say it is x
• Find the remainder when n is divided by x, say remainder r
• Find ar if r>0 and ax when r=0

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Tricks to Solve Questions on CAT Number system

Candidates should know and learn the basic purpose and concepts of numbers before learning the number system tricks and tips. There is a difference between natural numbers, real numbers, and integers. Try to understand what are these numbers.

• The last digit of a number in form: an
  Trick to know the last digit is to remember the cyclicity of the last digit of the given number. Each digit from 1 to 9 follow a pattern when exponential power is applied.
• To find out the number of zeroes
  The number of zeroes at the end of the factorial of at the end of the factorial of any number, divide the number by 5, the quotient attained is again divided by 5 and this is 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 the digits of the number, repeated until it becomes a single-digit number.
• Like, the digital root of 87983 is 8 + 7 + 9 + 8 + 3 ⇒ 35 = 3 + 5 ⇒ 8.
• The product of 3 consecutive natural numbers is completely divisible by 6.
• The product of 3 consecutive natural numbers, the first digit is an even number is perfectly divisible by 24.
• The sum of a two-digit number and a number formed by reversing the digits is perfectly divisible by 11.
  Like, 27 + 72 = 99, is divisible by 11.
• Also the difference between the above 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 + yn is 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.

Once, a candidate practices these questions in bulk on topics such as Unit Digit, the process of solving the question will be clear automatically.

CAT Samle Questions on Number System

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