Divisors of numbers

We have learned about factors of a number. Note that a number is a factor of a number n if and only if it is a divisor of n. In this section, we will learn about divisors of a number.

Let us now discuss what are divisors?

Definition of Divisors

A number $m$ is called a divisor of an integer $n$ if $\frac{m}{n}$ is an integer. In other words, $m$ is said to be a divisor of $n$ if $m$ divides $n$ completely with the remainder zero.

For example, $3$ is a divisor of $12$ as $\frac{12}{3}=4$ is an integer.

Similarly, both $4$ and $8$ are divisors of $16.$

 

Properties of Divisors

The divisors of a number have the following properties:

• The divisor of a number can be negative. Note that $\frac{12}{-3}=-4$ is an integer. Thus by the definition, $-3$ is a divisor of $12.$

• $1$ is always a divisor of any number.

• Any integer can be a divisor of $0.$

• A divisor of a number is always less than or equal to the number.

 

Formulas of Divisors

Suppose that the prime factorization of a number N is

N = pa × qb × rc

where p, q, and r are different prime numbers and a, b, c are natural numbers.

Number of divisors formula:

The total number of divisors of N is (a+1)(b+1)(c+1).

Sum of divisors formula:

The sum of divisors of N is equal to

$\frac{p^{a+1}-1}{p-1} \times \frac{q^{b+1}-1}{q-1}\times$ $\frac{r^{c+1}-1}{r-1}$

Product of divisors formula:

The product of divisors of N is equal to

NTotal No. of divisors/2

 

Table of Divisors 

Number N Divisors of N
1 1
2 1, 2
3 1, 3
Divisors of 4 1, 2, 4
5 1, 5
Divisors of 6 1, 2, 3, 6
7 1, 7
Divisors of 8 1, 2, 4, 8
Divisors of 9 1, 3, 9
Divisors of 10 1, 2, 5, 10
11 1, 11
Divisors of 12 1, 2, 3, 4, 6, 12
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
17  1, 17
Divisors of 18 1, 2, 3, 6, 9, 18
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
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
29  1, 29
Divisors of 30 1, 2, 3, 5, 6, 10, 15, 30
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
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
41  1, 41
Divisors of 42 1, 2, 3, 6, 7, 14, 21, 42
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
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
51 1, 3, 17, 51
52 1, 2, 4, 13, 26, 52
53  1, 53
Divisors of 54 1, 2, 3, 6, 9, 18, 27, 54
55  1, 5, 11, 55
Divisors of 56 1, 2, 4, 7, 8, 14, 28, 56
57 1, 3, 19, 57
58  1, 2, 29, 58
59  1, 59
Divisors of 60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
Divisors of 64 1, 2, 4, 8, 16, 32, 64
Divisors of 72 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
Divisors of 75 1, 3, 5, 15, 25, 75
Divisors of 100 1, 2, 4, 5, 10, 20, 25, 50, 100