This a continuation of article on number system. Here we will discuss few of the important concepts like divisibility, remainders HCF and LCM. This would help us in solving questions which require you to find the remainder when a large number is divided by another. Also, we'll discuss some shortcuts to find the remainders when a number is divided by specific numbers, quickly, based on the pattern.

## Objectives

- Rules of divisibility.
- LCM and HCF.
- Important properties of LCM and HCF.
- Remainders
- Properties of remainders.
- Concept of negative remainders.
- Pattern of remainders.
- Wilson's theorem about remainders.
- Numerical application of Remainder theorem.

## Rules of divisibility

In this area we are going to familiarize some interesting methods for checking the divisibility of some important natural numbers.

### Divisibility by 2 and 5

For checking the divisibility of 2 and 5, it is possible to apply the same method. If the unit place digit of a given number is divisible by 2, then the given number should be divisible by 2. Possible unit place digits are 0, 2, 4, 6, 8. Eg: 172, 234, 510, 2756, 1768 are divisible by 2.

Similarly, if the unit place digit of a number is divisible by 5, then the number is divisible by 5. Possible unit place digits are 0 and 5. Eg: 125, 5690 are divisible by 5.

### Divisibility by 4 and 25

If the last two digits (tens and unit place digits) are divisible by 4, then the number is divisible by 4. Similarly if the last two digits (tens and unit place digits) are divisible by 25, then the number is divisible by 25.

Eg: 100, 172, 8756, 5420 are divisible by 4.

1250, 3700, 54525, 7875 are divisible by 25.

### Divisibility by 8 and 125

For checking the divisibility of 8 and 125, it is enough to check the last three digits of the given number.

If the last three digits of a given number is divisible by 8 then the given number is divisible by 8 and similarly if the last three digits of a given number is divisible by 125 then the given number is divisible by 125.

Eg: 1960, 37648, 123488 are divisible by 8.

4500, 11125, 9375 are divisible by 125.

### Divisibility by 16 and 625

For checking the divisibility of 16 and 625, it is enough to check the last four digits of the given number.

If the last four digits of a given number is divisible by 16, then the given number is divisible by 16 and similarly if the last four digits of a given number is divisible by 625, then the given number is divisible by 625.

Eg: 357296, 83888, 1603760 are divisible by 16.

493125, 2283750 are divisible by 625.

### Divisibility by 3 and 9

For checking the divisibility of 3 and 9, check the sum of all digits in the number

If the sum of all digits in the given number is a multiple of 3, then the given number is divisible by 3.

Eg: 123, 3258, 152724 are divisible by 3.

If the sum of all digits in the given number is a multiple of 9 then the number is divisible by 9.

Eg: 12345678, 342711 are divisible by 9.

### Divisibility by 6,12, 14, 15, 18, 21, 44 and 108: Using co-prime factor method

For checking the divisibility of 6, 12, 14, 15, 18, 21 etc, check the given number is divisible by the co-prime factors of these divisors.

6 can be expressed as the product of its two co-prime factors in the form 2 x 3. So, for checking the divisibility of 6, check whether the given number is divisible by both 2 and 3.

Eg: 1452 is divisible by both 2 and 3, hence 1452 is divisible by 6.

Use the below table to check divisibility by 6,12, 14, 15, 18, 21, 44 and 108.

Divisor | Check the divisibility of |
---|---|

12 | 3 and 4 |

14 | 2 and 7 |

15 | 3 and 5 |

18 | 2 and 9 |

21 | 3 and 7 |

44 | 4 and 11 |

108 | 4 and 27 |

### Divisibility by 7, 13, 17, 19 and 23: Using operator and operation based method

To check divisibility by 7, 13, 17, 19 and 23, we can use operator and operation based method, for which below table would be handy. The process of divisibility check using operation and operator method is explained below.

Check the divisibility of | Operator | Operations |
---|---|---|

7 | 2 | Subtraction ( - ) |

13 | 4 | Addition ( + ) |

17 | 5 | Subtraction (-) |

19 | 2 | Addition ( + ) |

23 | 7 | Addition ( + ) |

Divisibility check using operation and operator method is done in three steps.

**Step 1:**Multiply the unit place of the given number by the mentioned operator**Step 2:**Apply the mentioned operation (subtraction or addition) on remaining part of the given number**Step 3:**Continue these steps till you can identify that the resultant difference is a multiple of the divisor or not. If this resultant difference is a multiple of divisor, then the given number is also a multiple of divisor, otherwise it is not.

`2214 - 2 xx 8 = 2198`

`219 - 2 xx 8 = 203`

`20 - 2 xx 3 = 14`, 14 it is a multiple of 7.

Hence 22148 is a multiple of 7.

`59381 + 4 xx 4 = 59397`

`5939 + 4 xx 7 = 5967`

`596 + 4 xx 7 = 624`

`62 + 4 xx 4 = 78`, 78 is divisible by 13.

Hence 593814 is a multiple of 13.

### Divisibility by 11: Using alternate digital sum method

If the difference between the sum of odd place digits and the sum of even place digits in a number is divisible by 11 (either '0' or any multiple of 11), then the given number should be a multiple of 11.

In the above illustration, difference between both the sums = 10 - 10 = 0. So 68123 is divisible by 11.