# Number Divisors

## Answer

## 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.

The tools listed at the bottom of this page ('See also' section) can be used to perform the divisor Finder Tools calculations.

### 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)
```

## See also

Divisibility Test

Euclidean division