Operations on Numbers

Face value and Place value

• For a given numeral Face value is the actual value of numeral for example for 6782 place value of 6 is 6,7 is 7,8 is 8 and so on.
• Place value at units place is calculated as
• unit digit X 1 e.g. 2 in above number
• tens digit X 10 e.g. 8 in above number and so on

Types of Numbers

• Natural Numbers
• Start with 1 also called as Counting numbers 1,2,3,4,5,6
• Whole Numbers
• Start with 0 eg 0,1,2,3,4,5
• Integers
• All counting numbers including zeros,negatives and positives.
• Even and Odd Numbers
• Even Numbers : All numbers which are divisible by 2 are called as even numbers
• Odd Numbers : All Numbers not divisible by 2 are called as odd numbers.
• Prime Numbers
• A counting number is called a prime number if it has 2 factors itself and 1
• example 2,3,4,5,7,11,13,19 etc
• To test or check for a prime number find its perfect square root of nearest greater number then check whether prime numbers less than square root divide it or not.
• for example for 137 nearest perfect square root number is 144
• square root of 144 is 12.
• prime numbers till 12 are 1,3,5,7,11.
• Since 137 is not divisible by either of them except 1 it is a prime number.
• Composite Numbers
• Natural numbers which are not prime are called as composite numbers.
• Coprimes
• coprime numbers have their HCF=1

Test of Divisibility

• By 2
• No which has even unit digits are divisible by 2
• By 3
• If sum of digits is divisible by 3 then number is divisible by 3
• By 9
• If sum of digits is divisible by 9 then number is divisible by 9
• By 4
• If the sum of its last two digits is divisible by 4
• By 8
• If the number formed by Hundred,Tens and Units digit of the given number is divisible by 8.
• By 10
• A number is divisible by 10 only when its unit digit is 0.
• By 5
• A number is divisible by 5 when its last digit is 0 or 5.
• By 11
• A number is divisible by 11 if the difference between sum of its digits at odd places and sum of its digits at even places is either 0 or difference is divisible by 11.

Series Formula

1+2+3+4+....+n=1/2*n*(n+1)

(1 pow 2 + 2 pow 2 + 3 pow 2+... n pow 2) = 1/6*n*(n+1)*(2n+1)

(1 pow 3 + 2 pow 3 + 3 pow 3 +......n pow 3)= 1/4*n pow 2 *(n+1) pow 2

Arithmetic Progression

a, a+d, a+2d, a+3d... are in arithmetic progression where first term is "a" and common difference is "d".

nth term = a+(n-1)d
sum of n terms = n/2*(2*a+(n-1)*d)
sum of n terms=n/2*(a+l)
where l is the last term

Geometric Progression

a,a*r,a*r pow 2, a*r pow 3.... are in Geometric progression if the first term = "a" and common ratio ="r"
sum of n terms is

a(1-r pow n)/(1-r) when r<1
a(r pow n -1)/(r-1) when r>1