Number Theory

Types of Numbers 
1. Natural Numbers
    Counting numbers 1,2,3,4,5,..........are called natural numbers
2.Whole Numbers

    All counting numbers together with zero form the set of whole numbers   i.e.  0,1,2,3,4,..........

3. Integers

   All natural numbers, 0 and negatives of counting numbers together form the set of integers.

   i.e. .....$...,-3,-2,-1,0,1,2,3,....<span\; style="font-weight:\; 500;">.etc.</span>$ 

4. Even Numbers

  A number divisible by 2 is called an even number

 i.e. .....$...,-6,-4,-2,0,2,4,6,....<span\; style="font-weight:\; 500;">.etc.</span>$

$<span\; style="color:\; blue;">5.Odd\; Numbers</span>$

  A number not divisible by 2 is called an odd number.

   i.e.,1,3,5,7,9,11, ..........etc.

6.Prime Numbers

   A number greater than 1 is called a prime number, if it has exactly two factors, 
   namely 1 and the   

    number itself.

    i.e. 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53 etc.

7.Composite Numbers

    Numbers greater than 1 which are not prime, are known as composite numbers,

 i.e. 4,6,8,9,10,12,.........etc.

8. Rational Numbers
     The number which is dividing one integer by another integer but not dividing by zero 
     is rational number 
      i.e.p/q,q is not zero     
      Example:- 2/3,8/7 etc.
9.Irrational Number
  Any real number which is not rational is irrational number.
   i.e.  e ,square root of 2,π,√3.
10.Real Numbers
     All rational and irrational numbers are real numbers it can be positive, negative ,zero
11. Imaginary Numbers
      A number when square give negative number.
      As √-9 = 3i, where i² = -1
12.Complex Number
     A combination of real number and imaginary number is a complex number.
      as 1+i,2-6i.
Divisibility Test :-
(1) Divisibility by 2
      If the last digit of any number is 0,2,4,6,8 or last digit is divisible by 2
      then number is divisible by 2.
       As 24,368,450,3576.
(2) Divisibility by 3
      If the sum of all digits of any number is divisible by 3 then the number is divisible by 3
     As   354     Sum of all digits 3+5+4=12 is divisible by 3 hence 354 is divisible by 3.
(3) Divisibility by 4
      If the last  two digits of any number is divisible 4 then number is divisible by 4.
      As 312     last two digits 12 is divisible by 4 hence 312 is divisible by 4.
(4) Divisibility by 5
     If the last digit of any number is 0 or 5 then number is divisible by 5.
      As 315   last  digit  is 5 hence number is divisible by 5 .
(5) Divisibility by 6
     If the last digit of any number is 0,2,4,6,8 and sum of all digits is divisible by 3  then number
      is divisible by 6.
    As  702   last digit is 2 and sum of all digits is 9 (9 is divisible by 3)

(6) Divisibility by 7
      Remove the last digit,double it,subtract it from the remaining part o number (truncated original number)
      and continue doing this until one digit remains if this is 0 or 7 
    As  1449
            144-2(9)=126  Now 12-2(6) =0   hence number is divisible by 7.
(7) Divisibility by 8
      If the last  three digits of any number is divisible 8 then number is divisible by 8.
      As  14472    last three digits is 472 divisible by 8. hence 14472 is divisible by 4.
(8) Divisibility by 9
       If the sum of all digits of any number is divisible by 9 then the number is divisible by 9.

     As   6345     Sum of all digits 6+3+4+5 =18 is divisible by 9 hence 6345 is divisible by 9.

(9) Divisibility by 10

      If the last digit of any number is 0  then number is divisible by 10.

       As  6770  last digit is 0 hence number is divisible by 10.

(10) Divisibility by 11

       If difference of  the sum of all digits of odd place and the sum  all digits of even place  is 
      divisible by 11 then the  number is divisible by 11

      As 91531   sum of odd place digits is (9+5+1) - sum of odd place digits is (3+1) =11
        hence number is divisible by 11.
Some Important Results

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

     (2)  1²+2²+3²+........................n²=n(n+1)(2n+1)/6

    (3)  1³+2³+3³+........................n³=n²(n+1)²/4

Arithmetic Progression

       If a,a+d,a+2d,a+3d,........are said to be in A.P.in which first term is a 

      and common difference is d.

 (1)  n^th term =  a+(n-1)d

 (2)  Sum of n terms = n{2a+(n-1)d}/2

 (3)  Sum of n terms = n{a+l}/2,Where l is a last term.

Relatively Prime Integers

         Two integers a and b are relatively prime if 

         GCD(a, b) = 1.

Examples:-

         (1) 15 and 28 relatively prime.

               GCD(15, 28) = 1.

         (2) 55 and 28 relatively prime

                GCD(55, 28) = 1.

         (3) Are 35 and 28 relatively prime?

               GCD(35, 28) = 7.

              Hence 35,28 are not relative prime.