NUMBERS AND ALGEBRA GATE 2020 STUDY MATERIAL PDF CIVILENGGFORALL

Numbers
HCF and LCM
Progressions
Averages, Mean, Mode and Median
Algebraic Formulae and Their Application
Polynomials
Inequations
Quadratic Equation with Application to Inequalities

NUMBERS

Natural Numbers

All positive integers are natural numbers. e.g. 1, 2, 3, 4, 5,…… There are infinite natural numbers and number 1 is the least natural number.

Types of Natural Number s: Based on divisibility there are following two types of natural numbers:

1. Prime Numbers: A natural number larger than unity is a prime number if it does not have other divisors except itself and unity. Note: Unity i.e 1 is not a prime number.

To Check A Number is Prime or not:

(i) Take square root of the number.
(ii) Round of the square root to the next highest integer and call this number as Z.
(iii) Check for divisibility of the number N by all prime numbers below Z. If there is no number below the value of Z which divides N then the number will be prime. e.g. 241 is prime or not?

241 lies between 15 or 16.Hence take the value of Z =16. Prime numbers less than 16 are 2, 3, 5, 7, 11 and 13. 241 is not divisible by any of these. Hence we can conclude that 241 is a prime number.

2. Composite Numbers: The numbers which are not prime are known as composite numbers. Note: 1 is neither prime not composite

Co-Primes: Two numbers a and b are said to be co-primes, if their H.C.F is 1. e.g. (2, 3), (4, 5), (7, 9), (8, 11)…..

Tests for Divisibility

A number is divisible by

2, when its unit’s digit is even or 0.
3, when the sum of its digits is divisible by 3.
4 when number formed by last two right hand digits is divisible by 4, or if last two digits are 0’s.
5, when its unit’s digit is 5 or 0.
6, when it is divisible by 2 and 3 both.
8 when number formed by last three right hand digits is divisible by 8, or when last three digit are 0’s.

9, when sum of its digits is divisible by 9.
10, when its unit’s digit is 0.
11, when difference between sum of the digits in the odd and sum of the digits in the even places is 0 or a multiple of 11. Note: When any number with an even number of digits is added to its reverse, the sum is always a multiple of 11.
12, when it is divisible by 3 and 4 both.

HCF AND LCM Greatest Common Divisor (GCD) It is the greatest divisor common to two or more given numbers. It is also called HCF (Highest Common Factor).

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