What are the divisors of 3111?

1, 3, 17, 51, 61, 183, 1037, 3111

There is a total of 8 positive divisors.
The sum of these divisors is 4464.
The arithmetic mean is 558.

8 odd divisors

1, 3, 17, 51, 61, 183, 1037, 3111

How to compute the divisors of 3111?

A number N is said to be divisible by a number M (with M non-zero) if, when we divide N by M, the remainder of the division is zero.

$N mod M = 0$

Brute force algorithm

We could start by using a brute-force method which would involve dividing 3111 by each of the numbers from 1 to 3111 to determine which ones have a remainder equal to 0.

$Remainder = N - (M \times ⌊ \frac{N}{M} ⌋)$

(where $⌊ \frac{N}{M} ⌋$ is the integer part of the quotient)

3111 / 1 = 3111 (the remainder is 0, so 1 is a divisor of 3111)
3111 / 2 = 1555.5 (the remainder is 1, so 2 is not a divisor of 3111)
3111 / 3 = 1037 (the remainder is 0, so 3 is a divisor of 3111)
...
3111 / 3110 = 1.0003215434084 (the remainder is 1, so 3110 is not a divisor of 3111)
3111 / 3111 = 1 (the remainder is 0, so 3111 is a divisor of 3111)

Improved algorithm using square-root

However, there is another slightly better approach that reduces the number of iterations by testing only integers less than or equal to the square root of 3111 (i.e. 55.776339069537). Indeed, if a number N has a divisor D greater than its square root, then there is necessarily a smaller divisor d such that:

$D \times d = N$

(thus, if $\frac{N}{D} = d$ , then $\frac{N}{d} = D$ )

3111 / 1 = 3111 (the remainder is 0, so 1 and 3111 are divisors of 3111)
3111 / 2 = 1555.5 (the remainder is 1, so 2 is not a divisor of 3111)
3111 / 3 = 1037 (the remainder is 0, so 3 and 1037 are divisors of 3111)
...
3111 / 54 = 57.611111111111 (the remainder is 33, so 54 is not a divisor of 3111)
3111 / 55 = 56.563636363636 (the remainder is 31, so 55 is not a divisor of 3111)