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.
Table of Contents
Definition of Divisors
A number $m$ is called a divisor of an integer $n$ if $\dfrac{m}{n}$ 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 $\dfrac{12}{3}=4$ 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 $\dfrac{12}{-3}=-4$ 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
$\dfrac{p^{a+1}-1}{p-1} \times \dfrac{q^{b+1}-1}{q-1}\times$ $\dfrac{r^{c+1}-1}{r-1}$
Product of divisors formula:
The product of divisors of N is equal to
NTotal No. of divisors/2
Table of Divisors
| Number N | List of Divisors of N |
| Divisors 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 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 |
