What are the divisors of 565?

1, 5, 113, 565

There is a total of 4 positive divisors.
The sum of these divisors is 684.
The arithmetic mean is 171.

4 odd divisors

1, 5, 113, 565

How to compute the divisors of 565?

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 565 by each of the numbers from 1 to 565 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)

565 / 1 = 565 (the remainder is 0, so 1 is a divisor of 565)
565 / 2 = 282.5 (the remainder is 1, so 2 is not a divisor of 565)
565 / 3 = 188.33333333333 (the remainder is 1, so 3 is not a divisor of 565)
...
565 / 564 = 1.0017730496454 (the remainder is 1, so 564 is not a divisor of 565)
565 / 565 = 1 (the remainder is 0, so 565 is a divisor of 565)

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 565 (i.e. 23.769728648009). 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 $\frac{N}{d} = D$ )

565 / 1 = 565 (the remainder is 0, so 1 and 565 are divisors of 565)
565 / 2 = 282.5 (the remainder is 1, so 2 is not a divisor of 565)
565 / 3 = 188.33333333333 (the remainder is 1, so 3 is not a divisor of 565)
...
565 / 22 = 25.681818181818 (the remainder is 15, so 22 is not a divisor of 565)
565 / 23 = 24.565217391304 (the remainder is 13, so 23 is not a divisor of 565)