## About Simplification

Simplification is one of the most important part of Quantitative Aptitude section of any competitive exam. Today I am sharing all the techniques to solve Simplification questions quickly.

## Rules of Simplification

**V →**Vinculum

**B →**Remove Brackets – in the order ( ) , { }, [ ]

**O →**Of

**D →**Division

**M →**Multiplication

**A →**Addition

**S → **Subtraction

## Important Parts of Simplification

- Number System
- HCF & LCM
- Square & Cube
- Fractions & Decimals
- Surds & Indices

## Number System

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

## 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. |

## 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. |

## 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