What are the divisors of 4565?

1, 5, 11, 55, 83, 415, 913, 4565

There is a total of 8 positive divisors.
The sum of these divisors is 6048.
The arithmetic mean is 756.

8 odd divisors

1, 5, 11, 55, 83, 415, 913, 4565

How to compute the divisors of 4565?

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 4565 by each of the numbers from 1 to 4565 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)

4565 / 1 = 4565 (the remainder is 0, so 1 is a divisor of 4565)
4565 / 2 = 2282.5 (the remainder is 1, so 2 is not a divisor of 4565)
4565 / 3 = 1521.6666666667 (the remainder is 2, so 3 is not a divisor of 4565)
...
4565 / 4564 = 1.0002191060473 (the remainder is 1, so 4564 is not a divisor of 4565)
4565 / 4565 = 1 (the remainder is 0, so 4565 is a divisor of 4565)

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 4565 (i.e. 67.564783726435). 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 $\frac{N}{d} = D$ )

4565 / 1 = 4565 (the remainder is 0, so 1 and 4565 are divisors of 4565)
4565 / 2 = 2282.5 (the remainder is 1, so 2 is not a divisor of 4565)
4565 / 3 = 1521.6666666667 (the remainder is 2, so 3 is not a divisor of 4565)
...
4565 / 66 = 69.166666666667 (the remainder is 11, so 66 is not a divisor of 4565)
4565 / 67 = 68.134328358209 (the remainder is 9, so 67 is not a divisor of 4565)