# Number System

**Number System**

In the Quantitative Ability section of the TCS NQT (National Qualifier Test), the number system is an important topic that you might encounter. It includes different types of numbers and their properties. Here are some common concepts related to the number system that you may come across in the test:

**Natural Numbers: **The set of positive integers, including 1, 2, 3, 4, and so on.

**Whole Numbers:** The set of natural numbers along with zero, including 0, 1, 2, 3, and so on.

**Integers:** The set of positive and negative whole numbers, including ..., -3, -2, -1, 0, 1, 2, 3, ...

**Rational Numbers: **Numbers that can be expressed as a fraction p/q, where p and q are integers and q is not equal to zero. Rational numbers include both terminating decimals and repeating decimals.

**Irrational Numbers: **Numbers that cannot be expressed as a fraction and have non-repeating, non-terminating decimals, such as √2, √3, π, etc.

**Real Numbers:** It refers to a subset of numbers that includes all rational and irrational numbers. Real numbers are used extensively in various mathematical and quantitative concepts. They are represented on the number line, which includes both positive and negative numbers, as well as zero.

**Factors and Multiples: **

**Factors: **Factors are the numbers that divide a given number without leaving any remainder. In other words, it evenly divides the number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, as these numbers divide 12 without any remainder.

**Key points about factors:**

Every number has at least 2 factors, 1 and the number itself.

A prime number has only 2 factors: 1 and the number itself.

Factors are always integers.

**Multiples:**

Multiples are the numbers that result from multiplying a given number by any integer. In simple terms, multiples are obtained by multiplying the number with other whole numbers. For example, the multiples of 15 are 15, 30, 45, 60, and so on.

**Key points about multiples:**

Every number is a multiple of itself.

The multiples of any number are infinitely many.

Multiples are always integers.

**Relationship between Factors and Multiples:**

If a number 'X' is a factor of another number 'Y', then 'Y' is a multiple of 'X'.

For example, if 4 is a factor of 16, then 16 is a multiple of 4.

**Application in Quantitative Aptitude:**

Factors and multiples are used in various quantitative aptitude problems, including finding common factors and multiples, simplifying fractions, prime factorization, finding the greatest common divisor (GCD) or greatest common factor (GCF), finding the least common multiple (LCM), solving word problems, and more.

**Example:**

Let's find the factors and multiples of the number 20:

Factors of 25: 1, 5, 25

Multiples of 2: 2, 4, 6, 8, 10, ...

**Prime Numbers:** Numbers greater than 1 that have only two positive divisors: 1 and the number itself.

**Composite Numbers: **Numbers that have more than two positive divisors.

**LCM (Least Common Multiple) and GCD (Greatest Common Divisor): **LCM is the smallest multiple that two or more numbers have in common, while GCD is the largest number that divides 2 or more numbers without leaving a remainder.

Example 1:

Find the LCM of 15 and 25.

Solution:

Multiples of 15: 15, 30, 45, 60, ...

Multiples of 25: 25, 50, 75, 100, ...

The smallest multiple that is common to both 15 and 25 is 75. Hence, the LCM of 15 and 25 is 75.

Example 2:

Find the GCD of 15 and 25.

Solution:

Factors of 15: 1, 3, 5, 15

Factors of 25: 1, 5, 25

The greatest common factor that is common to both 15 and 25 is 5. Therefore, the GCD of 15 and 25 is 5.

**Divisibility Rules: **These are specific mathematical rules that determine whether one number is divisible by another without the need to perform the actual division operation. These rules are particularly useful in number theory and quantitative aptitude, as they simplify the process of finding factors, multiples, and prime numbers.

Examples:

- Divisibility by 2:

Example 1: Is 426 divisible by 2?

Solution: Yes, it is divisible by 2 because the last digit is 6, which is even.

Example 2: Is 315 divisible by 2?

Solution: No, it is not divisible by 2 because the last digit is 5, which is odd.

- Divisibility by 3:

Example 1: Is 738 divisible by 3?

Solution: Yes, it is divisible by 3 because the sum of its digits is 7 + 3 + 8 = 18, which is divisible by 3.

**Number Series: **Questions where you need to identify the pattern and find the missing number in a series of numbers.

**Arithmetic Series:**

An arithmetic series is a sequence of numbers where each term is obtained by adding a constant difference to the previous term.

Example: 2, 5, 8, 11, 14, ...

In this series, the common difference is 3 (5 - 2 = 3, 8 - 5 = 3, and so on).

**Geometric Series:**

A geometric series is a sequence of numbers where each term is obtained by multiplying the previous term by a constant ratio.

Example: 2, 6, 18, 54, 162, ...

In this series, the common ratio is 3 (6 / 2 = 3, 18 / 6 = 3, and so on).

**Fibonacci Series:**

The Fibonacci series is a sequence of numbers where each term is the sum of the two preceding terms.

Example: 0, 1, 1, 2, 3, 5, 8, 13, 21, ...

In this series, each number is the sum of the two previous numbers (0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, and so on).

**Square Numbers Series:**

A series of square numbers is a sequence of numbers where each term is the square of an integer.

Example: 1, 4, 9, 16, 25, 36, 49, 64, ...

Each number in this series is the square of its position (1^2 = 1, 2^2 = 4, 3^2 = 9, and so on).

**Cube Numbers Series:**

A series of cube numbers is a sequence of numbers where each term is the cube of an integer.

Example: 1, 8, 27, 64, 125, ...

Each number in this series is the cube of its position (1^3 = 1, 2^3 = 8, 3^3 = 27, and so on).

**Prime Numbers Series:**

A series of prime numbers is a sequence of numbers where each term is a prime number.

Example: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

Each number in this series is a prime number (numbers that have no divisors other than 1 and themselves).

These are just a few examples of number series. There are many more types of series, and each may follow a different pattern or rule. Identifying the pattern in a number series is a common problem-solving skill used in various quantitative aptitude tests and puzzles.

**Number Properties: **Questions related to even numbers, odd numbers, perfect squares, perfect cubes, etc.

Being familiar with these concepts and their applications will help you solve various number system-related problems that may appear in the Quantitative Ability section of the TCS NQT. Practise different types of questions to improve your understanding and problem-solving skills.