Day 2: Properties of Integers

Hello frndz,

today we will start arithmetic. I would like to start with Integers and their properties.


:- All of us know that integer is any number like {.....-3,-2,-1,0,1,2,3.......}. Suppose x and y are integers and in the form of y = xn. then x will be a divisor of y and y will be a multiple of x.
For example 28 = 7*4, in which 7 and 4 are factors(divisors) of 28 and 28 is a multiple of 7 and 4.


:- when you divide 28 by 8 then quotient is 3 and remainder is 4. So 28 = 8*3 + 4.
In general, if you divide y by x and quotient is q and remainder is r then y = x*q + r
when remainder is 0 then x is a factor of y because it completely divides y.
Got it.......

:- Every integer that is divisible by 2 is an even integer and every integer that is not divisible by 2 is a odd integer. So 0,2,4,6.... are even integers and 1,3,5,7.... are odd integers.
Even and Odd integers can be negative numbers also.


:- even * even = even | even * odd = even | odd * odd = odd
   even + even = even | even + odd = odd | odd + odd = even
   even - even = even | even - odd = odd | odd - odd = even
 
In general, when multiply, if one of the number is even then result will be even else Odd.
when Add or Subtract, if both are either even or odd then result will be even else Odd.


:- A prime number is a positive number that has exactly two different positive divisors, 1 and itself.
For eg. 2,3,5.....   1 is not a prime number because it has only one divisor 1.

Let express 81 as a product of prime factors.
81 = 9*9
     = 3*3*9 (9 can be further divide into 3*3)
     = 3*3*3*3 (same happen with second 9)

So 81 = 3^4 (3 is to power 4) or 3*3*3*3


:- Numbers in a row known as consecutive numbers. For eg. 0,1,2,3,4......
It can represented as n, n+1, n+2, n+3........ where n is an integers.

Now, 0,2,4,6,8.... are even consecutive numbers and can be represented as 2n, 2n+2, 2n+4........
(hint: 2n is an even number and every alternative number after this known as even consecutive numbers)

Same as, 1,3,5,7..... are odd consecutive numbers and can be represented as 2n+1, 2n+3, 2n+5.......


:- If any number n multiply or divide by 1 then result will be same 'n'.
1*n = n and n/1 = n

if any number n except 0 divide by itself, it always gives 1.
n/n = 1, for all except 0.


:- The integer 0 is neither positive nor negative. If any number n add or subtract with 0, it always gives n.
and if any number n multiply with 0, it always be 0.
So n + 0 = n ; n - 0 = n ; n * 0 = 0
The division by 0 is not defined. n/0 = not defined. 




Ok frndz, hope all of you have fun with Integers. Next day we will learn about Fractions.
Bye....


Er. Ankur Garg
garg.ankur6@gmail.com