What are the divisors of 2599?

1, 23, 113, 2599

There is a total of 4 positive divisors.
The sum of these divisors is 2736.
The arithmetic mean is 684.

4 odd divisors

1, 23, 113, 2599

How to compute the divisors of 2599?

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 2599 by each of the numbers from 1 to 2599 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)

2599 / 1 = 2599 (the remainder is 0, so 1 is a divisor of 2599)
2599 / 2 = 1299.5 (the remainder is 1, so 2 is not a divisor of 2599)
2599 / 3 = 866.33333333333 (the remainder is 1, so 3 is not a divisor of 2599)
...
2599 / 2598 = 1.0003849114704 (the remainder is 1, so 2598 is not a divisor of 2599)
2599 / 2599 = 1 (the remainder is 0, so 2599 is a divisor of 2599)

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 2599 (i.e. 50.980388386124). 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 $\frac{N}{d} = D$ )

2599 / 1 = 2599 (the remainder is 0, so 1 and 2599 are divisors of 2599)
2599 / 2 = 1299.5 (the remainder is 1, so 2 is not a divisor of 2599)
2599 / 3 = 866.33333333333 (the remainder is 1, so 3 is not a divisor of 2599)
...
2599 / 49 = 53.040816326531 (the remainder is 2, so 49 is not a divisor of 2599)
2599 / 50 = 51.98 (the remainder is 49, so 50 is not a divisor of 2599)