What are the divisors of 8231?

1, 8231

There is a total of 2 positive divisors.
The sum of these divisors is 8232.
The arithmetic mean is 4116.

2 odd divisors

1, 8231

How to compute the divisors of 8231?

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 8231 by each of the numbers from 1 to 8231 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)

8231 / 1 = 8231 (the remainder is 0, so 1 is a divisor of 8231)
8231 / 2 = 4115.5 (the remainder is 1, so 2 is not a divisor of 8231)
8231 / 3 = 2743.6666666667 (the remainder is 2, so 3 is not a divisor of 8231)
...
8231 / 8230 = 1.0001215066829 (the remainder is 1, so 8230 is not a divisor of 8231)
8231 / 8231 = 1 (the remainder is 0, so 8231 is a divisor of 8231)

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 8231 (i.e. 90.724858776413). 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 $\frac{N}{d} = D$ )

8231 / 1 = 8231 (the remainder is 0, so 1 and 8231 are divisors of 8231)
8231 / 2 = 4115.5 (the remainder is 1, so 2 is not a divisor of 8231)
8231 / 3 = 2743.6666666667 (the remainder is 2, so 3 is not a divisor of 8231)
...
8231 / 89 = 92.483146067416 (the remainder is 43, so 89 is not a divisor of 8231)
8231 / 90 = 91.455555555556 (the remainder is 41, so 90 is not a divisor of 8231)