Natural Numbers: {1,2,3,4,5,……..}
Whole Numbers: { 0,1,2,3,4,…….}
Set of Integers: {…….,-3,-2,-1,0,1,2,3,4,………}
Sum of First n Natural Numbers: n(n+1)/2
Sum of Square of First n Natural Numbers: n(n+1)(2n+1)/6
Sum of Cube of First n Natural Numbers: [n(n+1)/2]²
Even Numbers: {2,4,6,8,…..}
Odd Numbers: { 1,3,5,7,9,…….}
Sum of First n Even Numbers: n(n+1)
Sum of First n Odd Numbers: n²
Prime Numbers
Numbers Divisible by 1 & itself
Eg. 2,3,5,7,11,13,…….
Test to Decide Prime Numbers
Steps: For Number n
- Take whole number x such that x <= √n
- Take all Prime Numbers <= x
- If none of the above numbers divide n exactly then n is Prime else n is Nonprime.
Two numbers having only 1 as a common factor are called co-prime numbers.
Thus, 4 and 15 are co-prime numbers.
Note :
If a number is divisible by another number then it is divisible by each of the factors of that number.
If a number is divisible by two co-prime numbers then it is divisible by their product also.
If two given numbers are divisible by a number, then their sum is also divisible by that number.
If two given numbers are divisible by a number, then their difference is also divisible by that number.
Twin Prime Numbers: Two prime numbers which differs by 2 Eg. (3,5), (11,13), ……..
Composite Numbers: Numbers divisible by at least one number other than 1 Eg. 4,6,9,15,……
Rational Numbers: Expressed as p/q
Irrational Numbers:
- Non recurring & non terminating decimals
- can not be expressed in p/q form
- eg. √3, √5, …….
Note : Rational & Irrational numbers together make Real Numbers
Formula : Dividend = ( Divisor * Quotient) + Remainder
VBODMAS Rule :
V – Vernacular or Bar “¯”
B – Brackets Removing order (), {},[]
O – Of
D – Division
M – Multiplication
A – Addition
S – Subtraction
Tests for Divisibility of Numbers
Divisibility by 10 :
if a number has 0 in the ones place then it is divisible by 10.
eg. 10, 1030,15000,etc
Divisibility by 5 :
a number which has either 0 or 5 in its ones place is divisible by 5
eg. 25, 1020, etc
Divisibility by 2 :
a number is divisible by 2 if it has any of the digits 0, 2, 4, 6 or 8 in its ones place.
eg. 2410, 4356, 1358, 2972, 5974
Divisibility by 3 :
if the sum of the digits is a multiple of 3, then the number is divisible by 3.
eg. 21, 27, 36, 54, 219
Divisibility by 6 :
if a number is divisible by 2 and 3 both then it is divisible by 6 also.
eg. 18
Divisibility by 4 :
a number with 3 or more digits is divisible by 4 if the number formed by its last two digits (i.e. ones and tens) is divisible by 4.
eg. 212, 1936
Divisibility by 8 :
a number with 4 or more digits is divisible by 8, if the number formed by the last three digits is divisible by 8.
eg. 2112, 10240
Divisibility by 9 :
if the sum of the digits of a number is divisible by 9, then the number itself is divisible by 9.
eg. 981, 2754,
Divisibility by 11 :
find the difference between the sum of the digits at odd places (from the right) and the sum of the digits at even places (from the right) of the number. If the difference is either 0 or divisible by 11, then the number is divisible by 11.
