What are the divisors of 3585?

1, 3, 5, 15, 239, 717, 1195, 3585

There is a total of 8 positive divisors.
The sum of these divisors is 5760.
The arithmetic mean is 720.

8 odd divisors

1, 3, 5, 15, 239, 717, 1195, 3585

How to compute the divisors of 3585?

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 3585 by each of the numbers from 1 to 3585 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)

3585 / 1 = 3585 (the remainder is 0, so 1 is a divisor of 3585)
3585 / 2 = 1792.5 (the remainder is 1, so 2 is not a divisor of 3585)
3585 / 3 = 1195 (the remainder is 0, so 3 is a divisor of 3585)
...
3585 / 3584 = 1.0002790178571 (the remainder is 1, so 3584 is not a divisor of 3585)
3585 / 3585 = 1 (the remainder is 0, so 3585 is a divisor of 3585)

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 3585 (i.e. 59.874869519691). 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 $\frac{N}{d} = D$ )

3585 / 1 = 3585 (the remainder is 0, so 1 and 3585 are divisors of 3585)
3585 / 2 = 1792.5 (the remainder is 1, so 2 is not a divisor of 3585)
3585 / 3 = 1195 (the remainder is 0, so 3 and 1195 are divisors of 3585)
...
3585 / 58 = 61.810344827586 (the remainder is 47, so 58 is not a divisor of 3585)
3585 / 59 = 60.762711864407 (the remainder is 45, so 59 is not a divisor of 3585)