What are the divisors of 2109?

1, 3, 19, 37, 57, 111, 703, 2109

There is a total of 8 positive divisors.
The sum of these divisors is 3040.
The arithmetic mean is 380.

8 odd divisors

1, 3, 19, 37, 57, 111, 703, 2109

How to compute the divisors of 2109?

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 2109 by each of the numbers from 1 to 2109 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)

2109 / 1 = 2109 (the remainder is 0, so 1 is a divisor of 2109)
2109 / 2 = 1054.5 (the remainder is 1, so 2 is not a divisor of 2109)
2109 / 3 = 703 (the remainder is 0, so 3 is a divisor of 2109)
...
2109 / 2108 = 1.0004743833017 (the remainder is 1, so 2108 is not a divisor of 2109)
2109 / 2109 = 1 (the remainder is 0, so 2109 is a divisor of 2109)

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 2109 (i.e. 45.92385001282). 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 $\frac{N}{d} = D$ )

2109 / 1 = 2109 (the remainder is 0, so 1 and 2109 are divisors of 2109)
2109 / 2 = 1054.5 (the remainder is 1, so 2 is not a divisor of 2109)
2109 / 3 = 703 (the remainder is 0, so 3 and 703 are divisors of 2109)
...
2109 / 44 = 47.931818181818 (the remainder is 41, so 44 is not a divisor of 2109)
2109 / 45 = 46.866666666667 (the remainder is 39, so 45 is not a divisor of 2109)