What are the divisors of 9591?

1, 3, 23, 69, 139, 417, 3197, 9591

There is a total of 8 positive divisors.
The sum of these divisors is 13440.
The arithmetic mean is 1680.

8 odd divisors

1, 3, 23, 69, 139, 417, 3197, 9591

How to compute the divisors of 9591?

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 9591 by each of the numbers from 1 to 9591 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)

9591 / 1 = 9591 (the remainder is 0, so 1 is a divisor of 9591)
9591 / 2 = 4795.5 (the remainder is 1, so 2 is not a divisor of 9591)
9591 / 3 = 3197 (the remainder is 0, so 3 is a divisor of 9591)
...
9591 / 9590 = 1.0001042752868 (the remainder is 1, so 9590 is not a divisor of 9591)
9591 / 9591 = 1 (the remainder is 0, so 9591 is a divisor of 9591)

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 9591 (i.e. 97.933651009242). 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 $\frac{N}{d} = D$ )

9591 / 1 = 9591 (the remainder is 0, so 1 and 9591 are divisors of 9591)
9591 / 2 = 4795.5 (the remainder is 1, so 2 is not a divisor of 9591)
9591 / 3 = 3197 (the remainder is 0, so 3 and 3197 are divisors of 9591)
...
9591 / 96 = 99.90625 (the remainder is 87, so 96 is not a divisor of 9591)
9591 / 97 = 98.876288659794 (the remainder is 85, so 97 is not a divisor of 9591)