# Number Divisors

1, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210Do you have any suggestions to improve this page ?

This online calculator finds all the divisors of an integer. Example: 30 divisors are 1, 2, 3, 4, 5, 6, 10, 15 and 30.

## Find all divisors of a number

- Calculate the root (R) of the number
- Try the whole division of the number by integers lower than R i.e. 2, 3, 4... up to R
- Do not forget to include 1 and the number itself that are divisors

### Example : What are the divisors of 75 ?

\sqrt (75)\approx 8.66
The division of 75 by all integers between 2 and 8 is attempted and the divisor and quotient are retained when the remainder is equal to 0.
- 75 ÷ 2 = 37 remainder 1
- 75 ÷ 3 = 25 remainder 0 -> we retain 3 and 25
- 75 ÷ 4 = 18 remainder 3
- 75 ÷ 5 = 15 remainder 0 -> we retain 5 and 15
- 75 ÷ 6 = 12 remainder 3
- 75 ÷ 7 = 10 remainder 5
- 75 ÷ 8 = 9 remainder 3
We addi 1 and 75 to the list of already found divisors,
The divisors of 75 are 1, 3, 5, 15, 25 and 75.

### What are the divisors of 81 ?

\sqrt (81) = 9
The division of 81 is performed by all integers between 2 and 9, and the divisor and quotient are retained when the remainder is equal to 0.
- 81 ÷ 2 = 40 remaining 1
- 81 ÷ 3 = 27 remainder = 0 -> we retain 3 and 27
- 81 ÷ 4 = 20 remaining 1
- 81 ÷ 5 = 16 remaining 1
- 81 ÷ 6 = 13 remaining 3
- 81 ÷ 7 = 11 remainder 4
- 81 ÷ 8 = 10 remaining 1
- 81 ÷ 9 = 9 remainder 0 -> we retain 9
We add 1 and 81 to the list of divisors already found:
The divisors of 81 are 1, 3, 9, 27 and 81.

## Divisibility rules

To quickly find out if a number is divisible by 2, 3, 5, 9, etc, you can use the divisibility rules explained on this page: Divisibility rules

## List of divisors of numbers from 1 to 100

1: 1
2: 1,2
3: 1,3
4: 1,2,4
5: 1,5
6: 1,2,3,6
7: 1,7
8: 1,2,4,8
9: 1,3,9
10: 1,2,5,10
11: 1,11
12: 1,2,3,4,6,12
13: 1,13
14: 1,2,7,14
15: 1,3,5,15
16: 1,2,4,8,16
17: 1,17
18: 1,2,3,6,9,18
19: 1,19
20: 1,2,4,5,10,20
21: 1,3,7,21
22: 1,2,11,22
23: 1,23
24: 1,2,3,4,6,8,12,24
25: 1,5,25
26: 1,2,13,26
27: 1,3,9,27
28: 1,2,4,7,14,28 29: 1,29
30: 1,2,3,5,6,10,15,30
31: 1,31
32: 1,2,4,8,16,32
33: 1,3,11,33
34: 1,2,17,34
35: 1,5,7,35
36: 1,2,3,4,6,9,12,18,36
37: 1,37
38: 1,2,19,38
39: 1,3,13,39
40: 1,2,4,5,8,10,20,40
41: 1,41
42: 1,2,3,6,7,14,21,42
43: 1,43
44: 1,2,4,11,22,44
45: 1,3,5,9,15,45
46: 1,2,23,46
47: 1,47
48: 1,2,3,4,6,8,12,16,24,48
49: 1,7,49
50: 1,2,5,10,25,50
51: 1,3,17,51
52: 1,2,4,13,26,52
53: 1,53
54: 1,2,3,6,9,18,27,54
55: 1,5,11,55
56: 1,2,4,7,8,14,28,56
57: 1,3,19,57
58: 1,2,29,58
59: 1,59
60: 1,2,3,4,5,6,10,12,15,20,30,60
61: 1,61
62: 1,2,31,62
63: 1,3,7,9,21,63
64: 1,2,4,8,16,32,64
65: 1,5,13,65
66: 1,2,3,6,11,22,33,66
67: 1,67
68: 1,2,4,17,34,68
69: 1,3,23,69
70: 1,2,5,7,10,14,35,70
71: 1,71
72: 1,2,3,4,6,8,9,12,18,24,36,72
73: 1,73
74: 1,2,37,74
75: 1,3,5,15,25,75
76: 1,2,4,19,38,76
77: 1,7,11,77
78: 1,2,3,6,13,26,39,78
79: 1,79
80: 1,2,4,5,8,10,16,20,40,80
81: 1,3,9,27,81
82: 1,2,41,82
83: 1,83
84: 1,2,3,4,6,7,12,14,21,28,42,84
85: 1,5,17,85
86: 1,2,43,86
87: 1,3,29,87
88: 1,2,4,8,11,22,44,88
89: 1,89
90: 1,2,3,5,6,9,10,15,18,30,45,90
91: 1,7,13,91
92: 1,2,4,23,46,92
93: 1,3,31,93
94: 1,2,47,94
95: 1,5,19,95
96: 1,2,3,4,6,8,12,16,24,32,48,96
97: 1,97
98: 1,2,7,14,49,98
99: 1,3,9,11,33,99
100:1,2,4,5,10,20,25,50,100

## Programming

### Python

This python program finds all divisors of a given integer n.

- Notice that for each divisor i of n such as i <= sqrt (n) , there is a k divisor of n, greater than or equal to sqrt (n) such as,
i* k = n, k = n//i, n//i denotes in python the quotient of the Euclidean division of n by i.

- As a result, the search for divisors can be done among integers from 1 up to the integer immediately less than or equal to sqrt (n) . Other divisors greater than sqrt (n) can be deduced easily.

- The search for divisors is done in the loop [while (i <= n**0.5)], n**0.5 is (in python) 'n power half' i.e. root of n.

- Within the while loop, when we find a divisor i (n%i is then equal to 0, n%i is (in python) the remainder of the Euclidean division of n by i also called the modulo operator), then we store (by the append method) i and n//i in the divisor_list variable returned by the divisors function.

- We end up sorting the list of divisors with the sort() method.

def divisors (n):
divisors_list = []
i=1
while (i <= n**0.5):
if (n%i==0):
divisors_list.append(i)
p = n//i
if(i != p) :
divisors_list.append(p)
i=i+1

divisors_list.sort()
return(divisors_list)