## Number System

Number System is divided into 4 parts we are providing short tricks to remember all the parts of number system:

- Classification
- Divisibility Test
- Division& Remainder Rules
- Sum Rules

** TRICKS TO REMEMBER NUMBER SYSTEM :**

**1. Classification**

Types |
Description |

Natural Numbers: |
all counting numbers ( 1,2,3,4,5….∞) |

Whole Numbers: |
natural number + zero( 0,1,2,3,4,5…∞) |

Integers: |
All whole numbers including Negative number + Positive number(∞……-4,-3,-2,-1,0,1,2,3,4,5….∞) |

Even & Odd Numbers : |
All whole number divisible by 2 is Even (0,2,4,6,8,10,12…..∞) and which does not divide by 2 are Odd (1,3,5,7,9,11,13,15,17,19….∞) |

Prime Numbers: |
It can be positive or negative except 1, if the number is not divisible by any number except the number itself.(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61….∞) |

Composite Numbers: |
Natural numbers which are not prime |

Co-Prime: |
Two natural number a and b are said to be co-prime if their HCF is 1. |

**2. Divisibility**

Numbers |
IF A Number |
Examples |

Divisible by 2 |
End with 0,2,4,6,8 are divisible by 2 | 254,326,3546,4718 all are divisible by 2 |

Divisible by 3 |
Sum of its digits is divisible by 3 | 375,4251,78123 all are divisible by 3. [549=5+4+9][5+4+9=18]18 is divisible by 3 hence 549 is divisible by 3. |

Divisible by 4 |
Last two digit divisible by 4 | 5648 here last 2 digits are 48 which is divisible by 4 hence 5648 is also divisible by 4. |

Divisible by 5 |
Ends with 0 or 5 | 225 or 330 here last digit digit is 0 or 5 that mean both the numbers are divisible by 5. |

Divisible by 6 |
Divides by Both 2 & 3 | 4536 here last digit is 6 so it divisible by 2 & sum of its digit (like 4+5+3+6=18) is 18 which is divisible by 3.Hence 4536 is divisible by 6. |

Divisible by 8 |
Last 3 digit divide by 8 | 746848 here last 3 digit 848 is divisible by 8 hence 746848 is also divisible by 8. |

Divisible by 10 |
End with 0 | 220,450,1450,8450 all numbers has a last digit zero it means all are divisible by 10. |

Divisible by 11 |
[Sum of its digit in odd places-Sum of its digits in even places]= 0 or multiple of 11 |
Consider the number 39798847(Sum of its digits at odd places)-(Sum of its digits at even places)(7+8+9+9)-(4+8+7+3)(23-12)23-12=11, which is divisible by 11. So 39798847 is divisible by 11. |

Divisible by 12 |
[The number must be divisible by 3 and 4] | Consider the number 462157692 is divisible by 12 { last 2 digits 92, so divisible by 4, and sum 4+6+2+1+5+7+6+9+2 = 42 is divisible by 3} , So 462157692 is divisible by 12. |

Divisible by 13 |
[Multiply last digit with 4 and add it to remaining number in given number, result must be divisible by 13] | Consider the number 4568 is not divisible by 13 { 456 + (4*8) = 488 –> 48 + (4*8) = 80, 80 is not divisible by 13.} , So 4568 is not divisible by 13. |

Divisible by 14 |
[The number must be divisible by 2 and 7. Because 2 and 7 are prime factors of 14.] | |

Divisible by 15 |
[The number should be divisible by 3 and 5. Because 3 and 5 are prime factors of 15.] | |

Divisible by 16 |
[The number formed by last four digits in given number must be divisible by 16.] | 7852176 is divisible by 16 –> 2176 is divisible by 16. |

Divisible by 17 |
[Multiply last digit with 5 and subtract it from remaining number in given number, result must be divisible by 17] | |

Divisible by 18 |
[The number should be divisible by 2 and 9] | |

Divisible by 19 |
[Multiply last digit with 2 and add it to remaining number in given number, result must be divisible by 19] | |

Divisible by 20 |
[The number formed by last two digits in given number must be divisible by 20.] |

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**TRICKS **

**Generalized Divisibility rules :**

To check divisibility by a number You should check divisibility by highest power of each of its prime factors.

Remember that the divisibility by any two of the factors of a number is not sufficient to judge its divisibility.

**TRICKS ****1.** Suppose you are checking for **divisibility by 12**. **2,3,4,6 **are **factors** of **12**. You cannot say that the number is divisible by 12 by checking divisibility by only 2 and 4 or 4 and 6. **You should check divisibility by 3 and 4 to tell the divisibility by 12. Because 3 and 4 are prime factors of 12.**

**TRICKS ****2.** Now, we are checking **divisibility by 18**. Prime factors of 18 are {2 and 3²}. So, you should **check divisibility by 2 and 9 **. **don’t check **for just **2 and 3** and also don’t check for **3 and 6**.

**TRICKS **** 3 ****List of Prime Factors of numbers upto 100**

Basically prime numbers have factors 1 and the number itself. So, we can’t this rule apply for prime numbers. This is only for composite numbers.

**Divisibility by numbers which ends in 1,3,7,9**

So, this rule counts for prime numbers which have been missed in previous rule.

**TRICKS 4**

**To test divisibility by a number N which ends in 1,3,7,9 this method can be used.**

**Multiply N with any number to get 9 in the end. Add 1 to the result and divide it by 10.**

Store the above result as R.

We are checking whether a number X is divisible by N or not.

Split X as X = 10 y + z ;

X is divisible by N, only if Rz + y is divisible by N.

**EXAMPLE (A):
Find whether 645 is divisible by 23 or not**.

N =23 ; 23 * 3 = 69 ; so now N has 9 in the end.

R = (69 + 1) / 10 = 7 ;

X = 645 ; split X as X = 10 y + z ;

645 = (10 * 64) + 5 ;

Y = 64 ; z = 5 ;

Rz + y = (7 * 5) + 64 = 35 + 64 = 99 ;

99 is not divisible by 23. So, 645 is also not divisible by 23.

**EXAMPLE (B) :
Let us find 585 is divisible by 39 or not.**

N =39 ; so now N has 9 in the end.

R = (39 + 1) / 10 = 4 ;

X = 585 ; split X as X = 10 y + z ;

645 = (10 * 58) + 5 ;

Y = 58 ; z = 5 ;

Rz + y = (4 * 5) + 58 = 20 + 58 = 78 ;

78 is divisible by 39. So, 585 is also divisible by 39.

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**3. Division & Remainder Rules**

Suppose we divide 45 by 6

hence ,represent it as:

**dividend = ( divisor****✘****quotient ) + remainder**

**or**

**divisior= [(dividend)-(remainder] / quotient**

could be write it as

x = kq + r where (x = dividend,k = divisor,q = quotient,r = remainder)

**Example:**

**On dividing a certain number by 342, we get 47 as remainder. If the same number is divided by 18, what will be the remainder ?**

Number = 342k + 47

( 18 ✘19k ) + ( 18 ✘2 ) + 11

18 ✘( 19k + 2 ) +11.

Remainder = 11

**4. Sum Rules:**

(1+2+3+………+n) = ^{1}/_{2 }n(n+1)

(1^{2}+2^{2}+3^{2}+………+n^{2}) = ^{1}/_{6 }n (n+1) (2n+1)

(1^{3}+2^{3}+3^{3}+………+n^{3}) = ^{1}/4_{ }n^{2} (n+1)^{2}

**Arithmetic Progression (A.P.)**

a, a + d, a + 2d, a + 3d, ….are said to be in A.P. in which

a = First Term and d = Common Difference .

Let the nth term be t_{n} and last term = l, then

(a) nth term = a + ( n – 1 ) d

(b) Sum of n terms =^{n}/_{2}[2a + (n-1)d]

(c) Sum of n terms = ^{n}/_{2} (a+l) where l is the last term