What are the divisors of 3109?

1, 3109

There is a total of 2 positive divisors.
The sum of these divisors is 3110.
The arithmetic mean is 1555.

2 odd divisors

1, 3109

How to compute the divisors of 3109?

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 3109 by each of the numbers from 1 to 3109 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)

3109 / 1 = 3109 (the remainder is 0, so 1 is a divisor of 3109)
3109 / 2 = 1554.5 (the remainder is 1, so 2 is not a divisor of 3109)
3109 / 3 = 1036.3333333333 (the remainder is 1, so 3 is not a divisor of 3109)
...
3109 / 3108 = 1.0003217503218 (the remainder is 1, so 3108 is not a divisor of 3109)
3109 / 3109 = 1 (the remainder is 0, so 3109 is a divisor of 3109)

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 3109 (i.e. 55.758407437803). 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 $\frac{N}{d} = D$ )

3109 / 1 = 3109 (the remainder is 0, so 1 and 3109 are divisors of 3109)
3109 / 2 = 1554.5 (the remainder is 1, so 2 is not a divisor of 3109)
3109 / 3 = 1036.3333333333 (the remainder is 1, so 3 is not a divisor of 3109)
...
3109 / 54 = 57.574074074074 (the remainder is 31, so 54 is not a divisor of 3109)
3109 / 55 = 56.527272727273 (the remainder is 29, so 55 is not a divisor of 3109)