What are the divisors of 9585?

1, 3, 5, 9, 15, 27, 45, 71, 135, 213, 355, 639, 1065, 1917, 3195, 9585

There is a total of 16 positive divisors.
The sum of these divisors is 17280.
The arithmetic mean is 1080.

16 odd divisors

1, 3, 5, 9, 15, 27, 45, 71, 135, 213, 355, 639, 1065, 1917, 3195, 9585

How to compute the divisors of 9585?

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 9585 by each of the numbers from 1 to 9585 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)

9585 / 1 = 9585 (the remainder is 0, so 1 is a divisor of 9585)
9585 / 2 = 4792.5 (the remainder is 1, so 2 is not a divisor of 9585)
9585 / 3 = 3195 (the remainder is 0, so 3 is a divisor of 9585)
...
9585 / 9584 = 1.0001043405676 (the remainder is 1, so 9584 is not a divisor of 9585)
9585 / 9585 = 1 (the remainder is 0, so 9585 is a divisor of 9585)

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 9585 (i.e. 97.903013232484). 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 $\frac{N}{d} = D$ )

9585 / 1 = 9585 (the remainder is 0, so 1 and 9585 are divisors of 9585)
9585 / 2 = 4792.5 (the remainder is 1, so 2 is not a divisor of 9585)
9585 / 3 = 3195 (the remainder is 0, so 3 and 3195 are divisors of 9585)
...
9585 / 96 = 99.84375 (the remainder is 81, so 96 is not a divisor of 9585)
9585 / 97 = 98.814432989691 (the remainder is 79, so 97 is not a divisor of 9585)