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 $\frac{m}{n}$ 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 = p^{a }× q^{b} × r^{c}

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

N^{Total 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 |