What are the divisors of 2563?

1, 11, 233, 2563

There is a total of 4 positive divisors.
The sum of these divisors is 2808.
The arithmetic mean is 702.

4 odd divisors

1, 11, 233, 2563

How to compute the divisors of 2563?

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 2563 by each of the numbers from 1 to 2563 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)

2563 / 1 = 2563 (the remainder is 0, so 1 is a divisor of 2563)
2563 / 2 = 1281.5 (the remainder is 1, so 2 is not a divisor of 2563)
2563 / 3 = 854.33333333333 (the remainder is 1, so 3 is not a divisor of 2563)
...
2563 / 2562 = 1.0003903200625 (the remainder is 1, so 2562 is not a divisor of 2563)
2563 / 2563 = 1 (the remainder is 0, so 2563 is a divisor of 2563)

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 2563 (i.e. 50.626080235389). 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 $\frac{N}{d} = D$ )

2563 / 1 = 2563 (the remainder is 0, so 1 and 2563 are divisors of 2563)
2563 / 2 = 1281.5 (the remainder is 1, so 2 is not a divisor of 2563)
2563 / 3 = 854.33333333333 (the remainder is 1, so 3 is not a divisor of 2563)
...
2563 / 49 = 52.30612244898 (the remainder is 15, so 49 is not a divisor of 2563)
2563 / 50 = 51.26 (the remainder is 13, so 50 is not a divisor of 2563)