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Divisors of a Number

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

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Tag(s) : Arithmetics

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# Divisors of a Number

## Divisors of a Number Calculator

 I want to List all divisors of N Count how many divisors has N Calculate sum of divisors (excluding N) Calculate sum of divisors (including N)

## Check a divisor

### What is a divisor? (Definition)

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 $a$.

### How to calculate the divisors' list of a number N?

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.

### What is the list of divisors from 1 to 100?

NumberList of Divisors
Divisor of 11
Divisors of 21,2
Divisors of 31,3
Divisors of 41,2,4
Divisors of 51,5
Divisors of 61,2,3,6
Divisors of 71,7
Divisors of 81,2,4,8
Divisors of 91,3,9
Divisors of 101,2,5,10
Divisors of 111,11
Divisors of 121,2,3,4,6,12
Divisors of 131,13
Divisors of 141,2,7,14
Divisors of 151,3,5,15
Divisors of 161,2,4,8,16
Divisors of 171,17
Divisors of 181,2,3,6,9,18
Divisors of 191,19
Divisors of 201,2,4,5,10,20
Divisors of 211,3,7,21
Divisors of 221,2,11,22
Divisors of 231,23
Divisors of 241,2,3,4,6,8,12,24
Divisors of 251,5,25
Divisors of 261,2,13,26
Divisors of 271,3,9,27
Divisors of 281,2,4,7,14,28
Divisors of 291,29
Divisors of 301,2,3,5,6,10,15,30
Divisors of 311,31
Divisors of 321,2,4,8,16,32
Divisors of 331,3,11,33
Divisors of 341,2,17,34
Divisors of 351,5,7,35
Divisors of 361,2,3,4,6,9,12,18,36
Divisors of 371,37
Divisors of 381,2,19,38
Divisors of 391,3,13,39
Divisors of 401,2,4,5,8,10,20,40
Divisors of 411,41
Divisors of 421,2,3,6,7,14,21,42
Divisors of 431,43
Divisors of 441,2,4,11,22,44
Divisors of 451,3,5,9,15,45
Divisors of 461,2,23,46
Divisors of 471,47
Divisors of 481,2,3,4,6,8,12,16,24,48
Divisors of 491,7,49
Divisors of 501,2,5,10,25,50
Divisors of 511,3,17,51
Divisors of 521,2,4,13,26,52
Divisors of 531,53
Divisors of 541,2,3,6,9,18,27,54
Divisors of 551,5,11,55
Divisors of 561,2,4,7,8,14,28,56
Divisors of 571,3,19,57
Divisors of 581,2,29,58
Divisors of 591,59
Divisors of 601,2,3,4,5,6,10,12,15,20,30,60
Divisors of 611,61
Divisors of 621,2,31,62
Divisors of 631,3,7,9,21,63
Divisors of 641,2,4,8,16,32,64
Divisors of 651,5,13,65
Divisors of 661,2,3,6,11,22,33,66
Divisors of 671,67
Divisors of 681,2,4,17,34,68
Divisors of 691,3,23,69
Divisors of 701,2,5,7,10,14,35,70
Divisors of 711,71
Divisors of 721,2,3,4,6,8,9,12,18,24,36,72
Divisors of 731,73
Divisors of 741,2,37,74
Divisors of 751,3,5,15,25,75
Divisors of 761,2,4,19,38,76
Divisors of 771,7,11,77
Divisors of 781,2,3,6,13,26,39,78
Divisors of 791,79
Divisors of 801,2,4,5,8,10,16,20,40,80
Divisors of 811,3,9,27,81
Divisors of 821,2,41,82
Divisors of 831,83
Divisors of 841,2,3,4,6,7,12,14,21,28,42,84
Divisors of 851,5,17,85
Divisors of 861,2,43,86
Divisors of 871,3,29,87
Divisors of 881,2,4,8,11,22,44,88
Divisors of 891,89
Divisors of 901,2,3,5,6,9,10,15,18,30,45,90
Divisors of 911,7,13,91
Divisors of 921,2,4,23,46,92
Divisors of 931,3,31,93
Divisors of 941,2,47,94
Divisors of 951,5,19,95
Divisors of 961,2,3,4,6,8,12,16,24,32,48,96
Divisors of 971,97
Divisors of 981,2,7,14,49,98
Divisors of 991,3,9,11,33,99
Divisors of 1001,2,4,5,10,20,25,50,100

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

### What are divisibility criteria?

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 $0$.

### What are the numbers with exactly 2 divisors?

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

### What are the numbers with exactly 3 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}

### What are the numbers with exactly 5 divisors?

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

### What are divisors of zero (0)?

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 $0$).

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

### Which number is a divisor of all numbers?

The number 1 divides all numbers.

### What is a perfect number?

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...

### What is an abundant number?

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...

### What is a superabundant number?

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)

### What is a deficient number?

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...

### What are amicable numbers?

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

### How to find a number from its divisors?

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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