What are the divisors of 9631?

1, 9631

There is a total of 2 positive divisors.
The sum of these divisors is 9632.
The arithmetic mean is 4816.

2 odd divisors

1, 9631

How to compute the divisors of 9631?

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 9631 by each of the numbers from 1 to 9631 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)

9631 / 1 = 9631 (the remainder is 0, so 1 is a divisor of 9631)
9631 / 2 = 4815.5 (the remainder is 1, so 2 is not a divisor of 9631)
9631 / 3 = 3210.3333333333 (the remainder is 1, so 3 is not a divisor of 9631)
...
9631 / 9630 = 1.0001038421599 (the remainder is 1, so 9630 is not a divisor of 9631)
9631 / 9631 = 1 (the remainder is 0, so 9631 is a divisor of 9631)

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 9631 (i.e. 98.137658419182). 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 $\frac{N}{d} = D$ )

9631 / 1 = 9631 (the remainder is 0, so 1 and 9631 are divisors of 9631)
9631 / 2 = 4815.5 (the remainder is 1, so 2 is not a divisor of 9631)
9631 / 3 = 3210.3333333333 (the remainder is 1, so 3 is not a divisor of 9631)
...
9631 / 97 = 99.288659793814 (the remainder is 28, so 97 is not a divisor of 9631)
9631 / 98 = 98.275510204082 (the remainder is 27, so 98 is not a divisor of 9631)