What are the divisors of 4509?

1, 3, 9, 27, 167, 501, 1503, 4509

There is a total of 8 positive divisors.
The sum of these divisors is 6720.
The arithmetic mean is 840.

8 odd divisors

1, 3, 9, 27, 167, 501, 1503, 4509

How to compute the divisors of 4509?

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 4509 by each of the numbers from 1 to 4509 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)

4509 / 1 = 4509 (the remainder is 0, so 1 is a divisor of 4509)
4509 / 2 = 2254.5 (the remainder is 1, so 2 is not a divisor of 4509)
4509 / 3 = 1503 (the remainder is 0, so 3 is a divisor of 4509)
...
4509 / 4508 = 1.0002218278616 (the remainder is 1, so 4508 is not a divisor of 4509)
4509 / 4509 = 1 (the remainder is 0, so 4509 is a divisor of 4509)

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 4509 (i.e. 67.149087856798). 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 $\frac{N}{d} = D$ )

4509 / 1 = 4509 (the remainder is 0, so 1 and 4509 are divisors of 4509)
4509 / 2 = 2254.5 (the remainder is 1, so 2 is not a divisor of 4509)
4509 / 3 = 1503 (the remainder is 0, so 3 and 1503 are divisors of 4509)
...
4509 / 66 = 68.318181818182 (the remainder is 21, so 66 is not a divisor of 4509)
4509 / 67 = 67.298507462687 (the remainder is 20, so 67 is not a divisor of 4509)