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 |

**Another?**: use the form on top of this page to get the list of divisors of any other number.

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 $.

Note N the number,

Initialize the list of divisors

For i equal to 2 up to the root of N,

Attempt to divide N by i

If the remainder of the division is 0, then append i to the list of divisors

End for

Return list of divisors

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, etc.

__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 $).

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

A negative integer has the same divisors as its positive opposite. The divisors of $ \pm N $ are the divisors of $ N $.

__Example:__ Divisors of `-2` are the same as divisors of `2`.

Technically if $ d $ is a divisor of $ N $ then $ -d $ is also a divisor of $ N $, to avoid trivial repetition, negative divisors are ignored.

The number 1 divides all numbers.

By equivalence, all integer numbers are multiples of 1.

Definition: A perfect number is a natural number N whose 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, etc.

Definition: An abundant number is a natural number $ N $ whose 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, etc.

Definition: a superabundant number is a number that has 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 whose 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, etc.

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* on dCode.fr [online website], retrieved on 2023-10-01,

- 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 is the algorithm for finding the divisors of a number?
- 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)?
- What are divisors of a negative number?
- 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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