Number System

Sum of Numbers:

• 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
• Sum of first n odd numbers                               = n²
• Sum of first n even numbers                              = n(n+1)

Prime Numbers: 

Prime numbers upto 100 are : (25 Prime numbers upto 100)

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

• Every prime number greater than 3 can be written in the form of (6k+1) or (6k-1).

TRICK:

If number +1 or -1 is divisible by 6, then the number is a prime number.

Important points:

• π is an irrational number while 22/7 is not
• 1 is the smallest natural number
• 1 is neither prime nor composite
• 0 is neither positive nor negative. It is a neutral integer

Number of Divisors:

N is a composite number with 

N = (x^a)(y^b)(z^c) then no. of divisors = (a+1)(b+1)(c+1) 

where x,y,z are primes

Cyclicity: 

To find unit's place digit:

• numbers 0,1,5 and 6 repeat themselves i.e, unaltered by any power on them.
• numbers 2,3,7 and 8 have cyclicity of 4 i.e, they repeat themselves after every 4th power.
• 9 and 4 have cyclicity of two.

Rules for divisibility of numbers :

A number is divisible 

• by 2, if unit's place is even or zero
• by 3, if the sum of all digits is divisible by 3
• by 4, if last two digits are divisible by 4
• by 5, if digit at unit's place is 0 or 5
• by 6, if the number is divisible by both 2 and 3
• by 7, if twice of unit's digit minus the sum of other digits is either 0 or multiple of 7
• by 8, if last three digits are divisible by 8 or last three or more terms are 0
• by 9, if the sum of all digits is divisible by 9
• by 10, if unit's digit is 0
• by 11, if the sum of digits at odd places minus sum of digits at even places is either 0 or multiple of 11 
• by 12, if the number is divisible by both 3 and 4
• by 25, if last two digits are either 0 or multiple of 25

Important points :

• (xⁿ - aⁿ) is divisible by (x -a) for all values of n.
• (xⁿ - aⁿ) is divisible by (x + a) for even values of n.
• (xⁿ + aⁿ) is divisible by (x + a) for odd values of n.

Progression:

Arithmetic Progression (A.P) :

nth term  :         a + (n -1)d = L   where,

                       L = last term

                        a = first term

                        n = number of terms

                        d = difference between two consecutive terms

Sum   :           n/2[2a + (n - 1)d] = n/2(a + L)

Geometric Progression (G.P) : 

nth term       :                        arⁿ⁻¹   where, 

                                             r = common ratio

Sum of n terms   :                 [a(1 - rⁿ)]/(1 - r)       if  r < 1

                                              [a(rⁿ - 1)]/(r - 1)       if  r >1

Sum of infinite terms  :        a/(1 - r)

Harmonic Progression (H.P) : 

If there exists the common difference between the inverse of each term, then the progression is said to be harmonic progression.
e.g

1/4, 1/7, 1/10, 1/13,...............

Important Points : 

If a, b and c are in 

H.P, then b = 2ac/(a+c)

A.P, then b = (a+c)/2

G.P, then b =  √(ac)