What are the divisors of 2509?

1, 13, 193, 2509

There is a total of 4 positive divisors.
The sum of these divisors is 2716.
The arithmetic mean is 679.

4 odd divisors

1, 13, 193, 2509

How to compute the divisors of 2509?

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 2509 by each of the numbers from 1 to 2509 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)

2509 / 1 = 2509 (the remainder is 0, so 1 is a divisor of 2509)
2509 / 2 = 1254.5 (the remainder is 1, so 2 is not a divisor of 2509)
2509 / 3 = 836.33333333333 (the remainder is 1, so 3 is not a divisor of 2509)
...
2509 / 2508 = 1.0003987240829 (the remainder is 1, so 2508 is not a divisor of 2509)
2509 / 2509 = 1 (the remainder is 0, so 2509 is a divisor of 2509)

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 2509 (i.e. 50.089919145473). 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 $\frac{N}{d} = D$ )

2509 / 1 = 2509 (the remainder is 0, so 1 and 2509 are divisors of 2509)
2509 / 2 = 1254.5 (the remainder is 1, so 2 is not a divisor of 2509)
2509 / 3 = 836.33333333333 (the remainder is 1, so 3 is not a divisor of 2509)
...
2509 / 49 = 51.204081632653 (the remainder is 10, so 49 is not a divisor of 2509)
2509 / 50 = 50.18 (the remainder is 9, so 50 is not a divisor of 2509)