Complete guide to number system :-

**Number and Classification****Divisibility and Property****Remainder and it's Rules****Sum****Last Digit****Number of Zeroes at the end of the product****Binary Number**

### Number and Classification

#### Some Various Types

- Natural Numbers
- Whole Numbers
- Even Numbers
- Odd Numbers
- Prime Numbers
- Integers
- Rational Numbers
- Real Numbers
- Irrational Numbers

**[ Natural Numbers ]**

All counting numbers excluding numeric digit 'zero'.

Like : 1,2,3,4,5,6,7,8,9,10,11,........∞(till infinity).

**[ Whole Numbers ]**

All counting numbers including numeric digit 'zero'.

Like : 0,1,2,3,4,5,6,7,8,9,10,11,........∞

**[ Even Numbers ]**

The numbers which divisible by 2 those numbers are even numbers.

Like : 2,4,6,8,10,12,14,16,18,20,22,.......∞

**[ Odd Numbers ]**

The numbers which is not divisible by 2 those numbers are odd numbers.

Like : 1,3,5,7,9,11,13,15,17,19,21,23,25,.......∞

**[ Prime Numbers ]**

The numbers which is only divisible by itself those numbers are prime numbers.It has always only one factor which is 1.

Like : 1,2,3,5,7,11,13,17,19,23,29,31,37,41,43,.......∞

Note: In prime numbers , all are odd numbers except numeric digit 2.

=> Checking Whether a number is Prime or Not

The following are the steps to checking whether a number is prime or not.

**Step 1:**Let any number and check the nearest possible square root of the number by taking it's approximate value.

**Step 2:**Makes the list of all prime number series less than the resultant number.

**Step 3:**Divide each number of the series with the result. If it is exactly divide with any of these number then it is not a prime number otherwise it is.

**Example :**

Check whether 239 is a prime number or not ?

**Sol:**

**[ Integers ]**

All positive and negative whole numbers are called integers .

Like : ∞........,-4,-3,-2,-1,0,1,2,3,4,.........∞

**[ Rational Numbers ]**

All fraction are rational numbers.

Like : 1/2,1/3,3/5

**[ Real Numbers ]**

A number could be positive,negative or fraction arranged at a interval are real number .

Like : ∞.......-3/2,-1,-1/2,0,1/2,1,3/2.....∞

**[ Irrational numbers ]**

The number which cannot be represented as a fraction.

Like : Π , √3, √2, √5

### Divisibility and Property

Numbers | Divisibility Test |

2 | All even number(including 0) is divisible by 2 |

3 | If the sum of digit of a number is divisible by 3 then the number is also divisible by 3. |

4 | If the last two digit of the number is divisible by 4 then the number is also divisible by 4(like 400). |

5 | If the number ends with 5 or 0. |

6 | If the number exactly divides by 2 and 3 then it also divisible by 6. |

8 | If the last three digit of the number is divisible by 8 then the number is also divisible by 8.(like 4ooo) |

9 | If The sum of the digit of number divisible by 9 then the number is also divisible by 9. |

**Properties of Divisibility**

- If a number A is divisible by B then the factors of A is also divisible by B.
- If a number A divides p then the sum of digits of p is also divisible by B.
- If a number A divides p then the difference of digits of p is also divisible by B.

### Remainder and its rule

If a number A divide by B then what would happen. Either it leaves remainder or not(means completely divides).

**Case 1:**(leaves remainder)

dividend = divisor × quotient + remainder

A = B × Q + R

**Case 2.**(leaves no remainder )

dividend = divisor × quotient

A = BQ

**Making a number completely or exactly divisible to given number**

There is a two way if we want that the given number should not leaves any remainder and get exactly divides with the divisor or any number.we have two method

**Method 1:**Subtracting the remainder with the given number or dividend . It is also used in finding the greatest n digit no exactly divisible by the given number.

**Method 2:**Subtract the divisor from remainder and then add with dividend.It is also used in finding the lowest n digit no exactly divisible by the given number.

Example:

Not to make 1326 exactly divisible with 12 we have two method by following:

**Method (1):**

1326-6 = 1320

**Method (2):**

1326+12-6 = 1332

**Finding Greatest n digit number exactly divisible by the given number**

Sol:

Applying Method (1) :

9999-24 = 9975

**Find the least 4 digit number exactly divisible by 28**

**Sol:**

1000 + 28 - 20 = 1008

**Sum**

**Find the sum of all natural number from 1 to 50 ?**

**Sol:**

**1+2+3+4+..........+50 =**

^{n(n+1)}/_{2}50×51/2 = 2550/2 = 1275

**Find the sum of even numbers from 2 to 10 ?**

**Sol:**

**2+4+6+8+10 = n(n+1)**

5×6 = 30

**Find the sum of odd number from 1 to 17 ?**

**Sol:**

**1+3+5+7+9+11+13+15+17 = (n)**

^{2}(9)

^{2}= 81

**Find the sum of squares from 1 to 18 ?**

**Sol:**

**(1)**

^{2}+(2)^{2}+(3)^{2}+(4)^{2}^{.....}+(18)^{2 }=^{n(n+1)(2n+1)}/_{6}18×19×37/6 = 12654/6 = 2109

**Find the sum of cubes from 1 to 12 ?**

**Sol:**

**(1)**

^{3}+(2)^{3}+(3)^{3}+(4)^{3}^{.....}+(12)^{3 }= [n(n+1)/2]^{2}[12×13/2]

^{2}= 156/2 = (78)

^{2}= 5084

### Last Digit

Following Table will help you in finding a last digit for any number.Just Let the last digit of the number as a Z.

**Find the last digit of (20373)**

^{130}?**Sol:**

(20373)

^{n}, here n = 130 and Z = 3130 divide by 4 leaves remainder = 2

the last digit for (20373)

which means 3^{130}= Z^{2}^{2 }= 9.

**Find the last digit**

^{ }(5548)^{228 }?**Sol:**

here n = 228, n divide by 4.

228/4 leaves remainder = 4

last digit (8)

^{4}= 4096which means 6

### Number of Zeroes at the end of the Product

To find the number of zeroes at the end of the product we need to know the number of squares of 2 and 5 in a multiplication.

(5×9) + (5×3) + (21×2) +(2)

No. of 2's(only 2) = 2

no. of 2's = 8

No. of 5's (only 5) = 5#### Example :

**Find the Number of zeroes at the end of :****45×15×42×32×14×22 ?****Sol:**(5×9) + (5×3) + (21×2) +(2)

^{5 }+ 2×7 + 2×11No. of 2's(only 2) = 2

^{1+5+1+1}no. of 2's = 8

^{1+1}

no. of 5's = 2

So pick whichever has contains the least number of digit from 2 and 5 only

no. of 2's = 8

no. of 5's = 2

because here in this case is 2 hence number of zeroes at the end is two.

### Binary Number

The number which we knew is 0,1,2,3,4,5,6,7,8,9. this number system is known as decimal number which contains 10 digits from zero to ten therefor it has base 10. If we already have a number system which is decimal number system so why we need binary number because computer only understand either 1 or 0.

As in decimal number we perform mathematical operation like subtraction , addition and multiplication same happens in case of binary number.

First of all we understand how to convert a decimal number into binary number and binary number into decimal number.

Binary number has its base 2 because it has only two digits 0 and 1. and the decimal number has its base 10 because it has only ten digits from 0 to 9.

**Conversion of decimal number into binary number**

**Convert (212)**

_{10}into ( ? )_{2}?**Sol:**

**Convert (101)**

^{2}into ( ? ) = ?**Sol:**

- Addition
- Subtraction
- Multiplication

**Addition**

Rule :0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0 carry = 1

**Subtraction **

1 - 0 = 11 - 1 = 0

0 - 0 = 0

0 - 1 = 1 carry = 1

**Multiplication**

0 × 0 = 00 × 1 = 0

1 × 0 = 0

1 × 1 = 1

**Example :**

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