What are the divisors of 4567?

1, 4567

There is a total of 2 positive divisors.
The sum of these divisors is 4568.
The arithmetic mean is 2284.

2 odd divisors

1, 4567

How to compute the divisors of 4567?

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 4567 by each of the numbers from 1 to 4567 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)

4567 / 1 = 4567 (the remainder is 0, so 1 is a divisor of 4567)
4567 / 2 = 2283.5 (the remainder is 1, so 2 is not a divisor of 4567)
4567 / 3 = 1522.3333333333 (the remainder is 1, so 3 is not a divisor of 4567)
...
4567 / 4566 = 1.0002190100745 (the remainder is 1, so 4566 is not a divisor of 4567)
4567 / 4567 = 1 (the remainder is 0, so 4567 is a divisor of 4567)

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 4567 (i.e. 67.579582715492). 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 $\frac{N}{d} = D$ )

4567 / 1 = 4567 (the remainder is 0, so 1 and 4567 are divisors of 4567)
4567 / 2 = 2283.5 (the remainder is 1, so 2 is not a divisor of 4567)
4567 / 3 = 1522.3333333333 (the remainder is 1, so 3 is not a divisor of 4567)
...
4567 / 66 = 69.19696969697 (the remainder is 13, so 66 is not a divisor of 4567)
4567 / 67 = 68.164179104478 (the remainder is 11, so 67 is not a divisor of 4567)