What are the divisors of 4540?

1, 2, 4, 5, 10, 20, 227, 454, 908, 1135, 2270, 4540

There is a total of 12 positive divisors.
The sum of these divisors is 9576.
The arithmetic mean is 798.

8 even divisors

2, 4, 10, 20, 454, 908, 2270, 4540

4 odd divisors

1, 5, 227, 1135

How to compute the divisors of 4540?

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 4540 by each of the numbers from 1 to 4540 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)

4540 / 1 = 4540 (the remainder is 0, so 1 is a divisor of 4540)
4540 / 2 = 2270 (the remainder is 0, so 2 is a divisor of 4540)
4540 / 3 = 1513.3333333333 (the remainder is 1, so 3 is not a divisor of 4540)
...
4540 / 4539 = 1.0002203128442 (the remainder is 1, so 4539 is not a divisor of 4540)
4540 / 4540 = 1 (the remainder is 0, so 4540 is a divisor of 4540)

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 4540 (i.e. 67.379522111692). 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 $\frac{N}{d} = D$ )

4540 / 1 = 4540 (the remainder is 0, so 1 and 4540 are divisors of 4540)
4540 / 2 = 2270 (the remainder is 0, so 2 and 2270 are divisors of 4540)
4540 / 3 = 1513.3333333333 (the remainder is 1, so 3 is not a divisor of 4540)
...
4540 / 66 = 68.787878787879 (the remainder is 52, so 66 is not a divisor of 4540)
4540 / 67 = 67.761194029851 (the remainder is 51, so 67 is not a divisor of 4540)