What are the divisors of 2539?

1, 2539

There is a total of 2 positive divisors.
The sum of these divisors is 2540.
The arithmetic mean is 1270.

2 odd divisors

1, 2539

How to compute the divisors of 2539?

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 2539 by each of the numbers from 1 to 2539 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)

2539 / 1 = 2539 (the remainder is 0, so 1 is a divisor of 2539)
2539 / 2 = 1269.5 (the remainder is 1, so 2 is not a divisor of 2539)
2539 / 3 = 846.33333333333 (the remainder is 1, so 3 is not a divisor of 2539)
...
2539 / 2538 = 1.0003940110323 (the remainder is 1, so 2538 is not a divisor of 2539)
2539 / 2539 = 1 (the remainder is 0, so 2539 is a divisor of 2539)

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 2539 (i.e. 50.388490749376). 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 $\frac{N}{d} = D$ )

2539 / 1 = 2539 (the remainder is 0, so 1 and 2539 are divisors of 2539)
2539 / 2 = 1269.5 (the remainder is 1, so 2 is not a divisor of 2539)
2539 / 3 = 846.33333333333 (the remainder is 1, so 3 is not a divisor of 2539)
...
2539 / 49 = 51.816326530612 (the remainder is 40, so 49 is not a divisor of 2539)
2539 / 50 = 50.78 (the remainder is 39, so 50 is not a divisor of 2539)