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

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 above to get the list of divisors of other numbers.

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

Example: 3^3 = 9, and 9 has three divisors 1,3,9

Example: 5^5 = 25, and 25 has three divisors 1,5,25

All integers divide zero, there is a infinite number of divisors of 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 is 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 and 284 are amicable :

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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- How to calculate the divisors' list of a number N?
- What is the list of divisors from 1 to 100?
- What are the numbers with exactly 3 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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