Tool to list divisors of a number. A divisor (or factor) of an integer number n is a number which divides n without remainder

Divisors of a Number - dCode

Tag(s) : Arithmetics

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The integer $ b $ (non-zero $ b \in \mathbb{N}_{>0} $) is a **divisor** of the integer $ a $ ($ \in \mathbb{N} $) if there is a integer $ c $ ($ \in \mathbb{N} $) such that $ c = a / b $ (NB: $ c $ is an integer, without decimal part).

In this case, $ c $ is represented as a division of $ a $ by $ b $ so $ b $ is indeed a **divisor** of $ a $ ($ a $ is divisible by $ b $).

By equivalence, $ a $ can be represented as a multiplication of $ b $ and $ c $: $ a = b \times c $, so $ a $ is a multiple of $ b $ and $ c $, and therefore $ b $ and $ c $ are **divisors** of $ a $.

An easy method consists in testing all numbers $ n $ between $ 1 $ and $ \sqrt{N} $ (square root of $ N $ ) to see if the remainder is equal to $ 0 $.

__Example:__ $ N = 10 $, $ \sqrt{10} \approx 3.1 $, $ 1 $ and $ 10 $ are always **divisors**, test $ 2 $: $ 10/2=5 $, so $ 2 $ and $ 5 $ are **divisors** of $ 10 $, test $ 3 $, $ 10/3 = 3 + 1/3 $, so $ 3 $ is not a **divisor** of $ 10 $.

Another method calculates the prime factors decomposition of $ N $ and by combination of them, get all **divisors**.

__Example:__ $ 10 = 2 \times 5 $, **divisors** are then $ 1 $, $ 2 $, $ 5 $, and $ 2 \times 5 = 10 $

Negative **divisors** also exist, but they are the same as positive **divisors** (with the sign close), so they are ignored.

Number | List of Divisors |
---|---|

Divisor of 1 | 1 |

Divisors of 2 | 1,2 |

Divisors of 3 | 1,3 |

Divisors of 4 | 1,2,4 |

Divisors of 5 | 1,5 |

Divisors of 6 | 1,2,3,6 |

Divisors of 7 | 1,7 |

Divisors of 8 | 1,2,4,8 |

Divisors of 9 | 1,3,9 |

Divisors of 10 | 1,2,5,10 |

Divisors of 11 | 1,11 |

Divisors of 12 | 1,2,3,4,6,12 |

Divisors of 13 | 1,13 |

Divisors of 14 | 1,2,7,14 |

Divisors of 15 | 1,3,5,15 |

Divisors of 16 | 1,2,4,8,16 |

Divisors of 17 | 1,17 |

Divisors of 18 | 1,2,3,6,9,18 |

Divisors of 19 | 1,19 |

Divisors of 20 | 1,2,4,5,10,20 |

Divisors of 21 | 1,3,7,21 |

Divisors of 22 | 1,2,11,22 |

Divisors of 23 | 1,23 |

Divisors of 24 | 1,2,3,4,6,8,12,24 |

Divisors of 25 | 1,5,25 |

Divisors of 26 | 1,2,13,26 |

Divisors of 27 | 1,3,9,27 |

Divisors of 28 | 1,2,4,7,14,28 |

Divisors of 29 | 1,29 |

Divisors of 30 | 1,2,3,5,6,10,15,30 |

Divisors of 31 | 1,31 |

Divisors of 32 | 1,2,4,8,16,32 |

Divisors of 33 | 1,3,11,33 |

Divisors of 34 | 1,2,17,34 |

Divisors of 35 | 1,5,7,35 |

Divisors of 36 | 1,2,3,4,6,9,12,18,36 |

Divisors of 37 | 1,37 |

Divisors of 38 | 1,2,19,38 |

Divisors of 39 | 1,3,13,39 |

Divisors of 40 | 1,2,4,5,8,10,20,40 |

Divisors of 41 | 1,41 |

Divisors of 42 | 1,2,3,6,7,14,21,42 |

Divisors of 43 | 1,43 |

Divisors of 44 | 1,2,4,11,22,44 |

Divisors of 45 | 1,3,5,9,15,45 |

Divisors of 46 | 1,2,23,46 |

Divisors of 47 | 1,47 |

Divisors of 48 | 1,2,3,4,6,8,12,16,24,48 |

Divisors of 49 | 1,7,49 |

Divisors of 50 | 1,2,5,10,25,50 |

Divisors of 51 | 1,3,17,51 |

Divisors of 52 | 1,2,4,13,26,52 |

Divisors of 53 | 1,53 |

Divisors of 54 | 1,2,3,6,9,18,27,54 |

Divisors of 55 | 1,5,11,55 |

Divisors of 56 | 1,2,4,7,8,14,28,56 |

Divisors of 57 | 1,3,19,57 |

Divisors of 58 | 1,2,29,58 |

Divisors of 59 | 1,59 |

Divisors of 60 | 1,2,3,4,5,6,10,12,15,20,30,60 |

Divisors of 61 | 1,61 |

Divisors of 62 | 1,2,31,62 |

Divisors of 63 | 1,3,7,9,21,63 |

Divisors of 64 | 1,2,4,8,16,32,64 |

Divisors of 65 | 1,5,13,65 |

Divisors of 66 | 1,2,3,6,11,22,33,66 |

Divisors of 67 | 1,67 |

Divisors of 68 | 1,2,4,17,34,68 |

Divisors of 69 | 1,3,23,69 |

Divisors of 70 | 1,2,5,7,10,14,35,70 |

Divisors of 71 | 1,71 |

Divisors of 72 | 1,2,3,4,6,8,9,12,18,24,36,72 |

Divisors of 73 | 1,73 |

Divisors of 74 | 1,2,37,74 |

Divisors of 75 | 1,3,5,15,25,75 |

Divisors of 76 | 1,2,4,19,38,76 |

Divisors of 77 | 1,7,11,77 |

Divisors of 78 | 1,2,3,6,13,26,39,78 |

Divisors of 79 | 1,79 |

Divisors of 80 | 1,2,4,5,8,10,16,20,40,80 |

Divisors of 81 | 1,3,9,27,81 |

Divisors of 82 | 1,2,41,82 |

Divisors of 83 | 1,83 |

Divisors of 84 | 1,2,3,4,6,7,12,14,21,28,42,84 |

Divisors of 85 | 1,5,17,85 |

Divisors of 86 | 1,2,43,86 |

Divisors of 87 | 1,3,29,87 |

Divisors of 88 | 1,2,4,8,11,22,44,88 |

Divisors of 89 | 1,89 |

Divisors of 90 | 1,2,3,5,6,9,10,15,18,30,45,90 |

Divisors of 91 | 1,7,13,91 |

Divisors of 92 | 1,2,4,23,46,92 |

Divisors of 93 | 1,3,31,93 |

Divisors of 94 | 1,2,47,94 |

Divisors of 95 | 1,5,19,95 |

Divisors of 96 | 1,2,3,4,6,8,12,16,24,32,48,96 |

Divisors of 97 | 1,97 |

Divisors of 98 | 1,2,7,14,49,98 |

Divisors of 99 | 1,3,9,11,33,99 |

Divisors of 100 | 1,2,4,5,10,20,25,50,100 |

Use the form on top of this page to get the list of **divisors** of other numbers.

The divisibility criteria are a roundabout way to know if a number is divisible by another without directly doing the calculation. Here is a (non-exhaustive) list of the main divisibility criteria (in base 10):

— Criterion of divisibility by $ 1 $: any integer number is divisible by $ 1 $

— Criterion of divisibility by $ 2 $: any number multiple of $ 2 $ has an even digit for the units digit, so the last digit is either $ 0 $ or $ 2 $ or $ 4 $ or $ 6 $ or $ 8 $.

— Criterion of divisibility by $ 3 $: any number multiple of $ 3 $ has for sum of digits a number which is also multiple of $ 3 $, and therefore the digital root of the number is $ 0 $ or $ 3 $ or $ $ 6 or $ 9 $

— Criterion of divisibility by $ 4 $: any number multiple of $ 4 $ has as the sum of the units digit and the double of the tens digit a number also divisible by 4. (Variant) the last 2 digits (tens and ones) of any number multiple of $ 4 $ are divisible by $ 4 $ (so by $ 2 $ then again by $ 2 $)

— Criterion of divisibility by $ 5 $: any number multiple of $ 5 $ has for digit of the units $ 0 $ or $ 5 $

— Criterion of divisibility by $ 6 $: any number multiple of $ 6 $ validates the criteria of divisibility by $ 2 $ and by $ 3 $

— Criterion of divisibility by $ 7 $: any number multiple of $ 7 $ has a sum of its total number of tens (all digits except the last) and of five times its units digit also divisible by 7 (criterion to be repeated in loop)

— Criterion of divisibility by $ 8 $: any number which is multiple of $ 8 $ has for the sum of the units digit, the double of the tens digit and the quadruple of the hundreds digit a number also divisible by 8.

— Criterion of divisibility by $ 9 $: any number which is multiple of $ 9 $ has as its sum a number which is also a multiple of $ 9 $, and therefore the digital root of the number is $ 9 $.

— Criterion of divisibility by $ 10 $: any number multiple of $ 10 $ has as last digit $ 0 $.

Numbers that have only 2 **divisors** are prime numbers. They have $ 1 $ and themselves as **divisors**.

Numbers having exactly 3 **divisors** are perfect squares of prime numbers: 4, 9, 25, 49, ...

__Example:__ 2^2 = 4, and 4 has three **divisors** {1,2,4}

3^2 = 9, and 9 has three **divisors** {1,3,9}

5^2 = 25, and 25 has three **divisors** {1,5,25}

Numbers having exactly 5 **divisor** are numbers of the form $ a^4 $ with $ a $ a prime number.

__Example:__ 2^4 = 16, and 16 has five **divisors** 1,2,4,8,16

3^4 = 81, and 81 has five **divisors** 1,3,9,27,81

The number $ 0 $ has an infinity of **divisors**, because all the numbers divide $ 0 $ and the result is worth $ 0 $ (except for $ 0 $ itself because the division by $ 0 $ does not make sense, it is however possible to say that $ 0 $ is a multiple of $ 0 $).

$$ \frac{0}{n} = 0, (n \neq 0) $$

The number 1 divides all numbers.

Definition: A perfect number is a natural number N which sum of **divisors** (excluding N) is equal to N.

__Example:__ $ 6 $ has for **divisors** $ 3 $, $ 2 $ and $ 1 $. And the sum $ 3+2+1=6 $, so $ 6 $ is a perfect number.

__Example:__ The first perfect numbers are: 6, 28, 496, 8128, 33550336, 8589869056, 137438691328...

Definition: An abundant number is a natural number $ N $ which sum of **divisors** (excluding $ N $) is superior to $ N $.

__Example:__ $ 12 $ has for **divisors** 6, 4, 3, 2 and 1. And the sum $ 6+4+3+2+1=15 $ superior to 12, so 12 is an abundant number.

__Example:__ The first abundant numbers are: 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100...

Definition: a superabundant number is a number that have more **divisors** than any number smaller than it.

__Example:__ $ 12 $ is superabundant because it has 6 **divisors**: 1,2,3,4,6,12 and no other smaller number has at least 6 **divisors**.

The first abundant numbers are: 1 (1 **divisor**), 2 (2 **divisors**), 4 (3 **divisors**), 6 (4 **divisors**), 12 (6 **divisors**), 24 (8 **divisors**), 36 (9 **divisors**), 48 (10 **divisors**), 60 (12 **divisors**), 120 (16 **divisors**), 180 (18 **divisors**), 240 (20 **divisors**), 360 (24 **divisors**), 720 (30 **divisors**), 840 (32 **divisors**), 1260 (36 **divisors**), 1680 (40 **divisors**), 2520 (48 **divisors**), 5040 (60 **divisors**), 10080 (72 **divisors**), 15120 (80 **divisors**), 25200 (90 **divisors**), 27720 (96 **divisors**), 55440 (120 **divisors**), 110880 (144 **divisors**), 166320 (160 **divisors**), 277200 (180 **divisors**), 332640 (192 **divisors**), 554400 (216 **divisors**), 665280 (224 **divisors**), 720720 (240 **divisors**), 1441440 (288 **divisors**), 2162160 (320 **divisors**), 3603600 (360 **divisors**), 4324320 (384 **divisors**), 7207200 (432 **divisors**), 8648640 (448 **divisors**), 10810800 (480 **divisors**), 21621600 (576 **divisors**)

Definition: A deficient number is a natural number N which sum of **divisors** (excluding N) is inferior to N.

__Example:__ $ 4 $ has for **divisors** 2 and 1. And 2+1=3 inferior to 4, so 4 is a deficient number.

__Example:__ The first deficient numbers are: 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50...

Two numbers are amicable if the sum of their **divisors** is the same and the sum of the two numbers is equal to the sum of their **divisors**.

__Example:__ 220 is amicable with 284 (they are amicable numbers) :

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 + 220 = 504

1 + 2 + 4 + 71 + 142 + 284 = 504

220 + 284 = 504

The least common multiple (LCM) is the smallest number that has for **divisors** a list of given numbers.

__Example:__ `2,4,10` has `20` for PPCM and thus `2`, `4` and `10` are **divisors** of `20`.

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- Divisors of a Number Calculator
- Check a divisor
- What is a divisor? (Definition)
- How to calculate the divisors' list of a number N?
- What is the list of divisors from 1 to 100?
- What are divisibility criteria?
- What are the numbers with exactly 2 divisors?
- What are the numbers with exactly 3 divisors?
- What are the numbers with exactly 5 divisors?
- What are divisors of zero (0)?
- Which number is a divisor of all numbers?
- What is a perfect number?
- What is an abundant number?
- What is a superabundant number?
- What is a deficient number?
- What are amicable numbers?
- How to find a number from its divisors?

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