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The Number Tree


Important Concepts: Number System - 2 | CSAT Preparation - UPSC

10. Odd and Even Numbers

Important Concepts: Number System - 2 | CSAT Preparation - UPSC

  • This is a property of Integers. But since our questions are limited to the Whole numbers, we will keep our studies limited to them.
  • By Definition, Even Numbers are those which are divisible by 2. They are the numbers with their unit digit as either 2, 4, 6, 8 or 0. The general expression of Even numbers is 2n where n is a whole number.
  • By Definition, Odd Numbers are those which are not divisible by 2. They are the numbers with their unit digit as either 1, 3, 5, 7 or 9. The general expression of Odd numbers is 2n+1 where n is a whole number.

Certain Operations on Whole Numbers:
(i) Odd ± Odd = Even
(ii) Even ± Even = Even
(iii) Odd ± Even = Odd
(iv) Even ± Even ± Even ........... n terms = Even
(v) Odd ± Odd ± Odd .............. n terms = Even if n is Even, Odd if n is Odd
(vi) Odd × Odd = Odd
(vii) Even × Odd = Even
(viii) Even × Even = Even
(ix) Even × Even × Even ....... n terms = Even
(x) Odd × Odd × Odd ....... n terms = Odd
(xi) (Even)n = Even
(xii) (odd)n = Odd

Question for Important Concepts: Number System - 2
Try yourself:If a is even what would 5a + 1 be
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Question for Important Concepts: Number System - 2
Try yourself:If a is even and b is odd what would a3 + 3b + 6 be
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Question for Important Concepts: Number System - 2
Try yourself:If 5a - 2 is even what would a  be
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11. Prime and Composite Numbers


(a) Prime Numbers: Being Prime is a property of Natural Numbers. All those numbers which are divisible by exactly 2 numbers, i.e. itself and unity "1".
Example: 2, 3, 5, 7, 11 etc. 

(b) Composite are the numbers which are divisible by more than 2 numbers, i.e. itself, unity "1" and at least one other.
Example: 4, 6, 8, 9, 10 etc. 

1 is neither Prime nor Composite. 

Important Concepts: Number System - 2 | CSAT Preparation - UPSC

Question for Important Concepts: Number System - 2
Try yourself:How many Prime numbers are there less than 100
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Question for Important Concepts: Number System - 2
Try yourself:Which of the following cannot be the sum of two prime numbers. 
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Question for Important Concepts: Number System - 2
Try yourself:State Whether True or False 

All Prime numbers are Odd

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Question for Important Concepts: Number System - 2
Try yourself:State Whether True or False 

Sum of Any 2 prime numbers is always Even

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Question for Important Concepts: Number System - 2
Try yourself:State Whether True or False 

Product of Any two prime numbers is always Odd

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Question for Important Concepts: Number System - 2
Try yourself:State Whether True or False 

There are no 4 digit Prime numbers.

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Property of Prime Numbers

  • All Prime numbers greater than 3 are of the form 6n ± 1 but not all numbers of form 6n ± 1 are Prime. This property means when prime numbers greater than 3 are divided by 6 they leave a remainder of 1 or 5.

Question for Important Concepts: Number System - 2
Try yourself:Which of the following is a prime numbers. 
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12. Co - Prime Numbers


  • This is a property of a pair or more than two natural numbers. 
  • Two numbers are said to be co-prime if the only number which can completely divide both the numbers is 1. No number other than 1 can divide both the given numbers of the pair. 
  • These numbers in themselves can be prime or composite. 1 is co-prime with all the natural numbers other than 1.
    Example: (3, 7); (4, 15); (3, 10) etc. 
  • A set of numbers is said to be co-prime if their HCF is 1. We will be learning about HCF in the following chapters.
    Example: (3, 7, 11); (4, 15, 10); (3, 10, 8) etc.
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FAQs on Important Concepts: Number System - 2 - CSAT Preparation - UPSC

1. What is the number system?
Ans. The number system is a mathematical system that represents numbers and their relationships. It includes various types of numbers, such as natural numbers, whole numbers, integers, rational numbers, and irrational numbers. These numbers are used to quantify and measure quantities or values.
2. What are the different types of numbers in the number system?
Ans. The number system consists of several types of numbers. Here are some of the main types: - Natural numbers: These are the counting numbers starting from 1 and going infinitely. - Whole numbers: These include all natural numbers along with the number 0. - Integers: Integers include both positive and negative whole numbers, along with zero. - Rational numbers: Rational numbers are numbers that can be expressed as a fraction or a quotient of two integers. - Irrational numbers: Irrational numbers cannot be expressed as a fraction and have an infinite number of non-recurring decimal places.
3. What are the properties of numbers in the number system?
Ans. Numbers in the number system possess various properties. Some of the important properties include: - Commutative property: The order of numbers can be changed without affecting the result of addition or multiplication. For example, a + b = b + a. - Associative property: The grouping of numbers can be changed without affecting the result of addition or multiplication. For example, (a + b) + c = a + (b + c). - Distributive property: Multiplication can be distributed over addition or subtraction. For example, a × (b + c) = (a × b) + (a × c). - Identity property: The sum of any number and zero is equal to the number itself. Similarly, the product of any number and one is equal to the number itself. - Inverse property: Every number has an additive inverse (opposite) and a multiplicative inverse (reciprocal) except for zero.
4. How is the number system used in everyday life?
Ans. The number system is used extensively in everyday life for various purposes, such as: - Counting: We use numbers for counting objects, people, money, and many other things. - Measurements: Numbers are used to quantify measurements of length, weight, time, temperature, and other physical quantities. - Money and finance: Numbers are used for calculating expenses, budgets, investments, and financial transactions. - Science and technology: Numbers are fundamental in scientific calculations, engineering, computer programming, and data analysis. - Sports and games: Numbers are used for scoring, statistics, rankings, and odds calculations in sports and games.
5. How does the number system relate to other branches of mathematics?
Ans. The number system serves as the foundation for various branches of mathematics. It provides the basis for operations like addition, subtraction, multiplication, and division, which are essential in algebra, calculus, and other areas of mathematics. The properties and relationships between different types of numbers in the number system form the building blocks for more advanced mathematical concepts. Additionally, the number system is intertwined with topics like equations, inequalities, fractions, decimals, and percentages, which are used extensively in many mathematical disciplines.
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