What are the divisors of 2691?

1, 3, 9, 13, 23, 39, 69, 117, 207, 299, 897, 2691

There is a total of 12 positive divisors.
The sum of these divisors is 4368.
The arithmetic mean is 364.

12 odd divisors

1, 3, 9, 13, 23, 39, 69, 117, 207, 299, 897, 2691

How to compute the divisors of 2691?

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 2691 by each of the numbers from 1 to 2691 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)

2691 / 1 = 2691 (the remainder is 0, so 1 is a divisor of 2691)
2691 / 2 = 1345.5 (the remainder is 1, so 2 is not a divisor of 2691)
2691 / 3 = 897 (the remainder is 0, so 3 is a divisor of 2691)
...
2691 / 2690 = 1.0003717472119 (the remainder is 1, so 2690 is not a divisor of 2691)
2691 / 2691 = 1 (the remainder is 0, so 2691 is a divisor of 2691)

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 2691 (i.e. 51.874849397372). 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 $\frac{N}{d} = D$ )

2691 / 1 = 2691 (the remainder is 0, so 1 and 2691 are divisors of 2691)
2691 / 2 = 1345.5 (the remainder is 1, so 2 is not a divisor of 2691)
2691 / 3 = 897 (the remainder is 0, so 3 and 897 are divisors of 2691)
...
2691 / 50 = 53.82 (the remainder is 41, so 50 is not a divisor of 2691)
2691 / 51 = 52.764705882353 (the remainder is 39, so 51 is not a divisor of 2691)