What are the divisors of 8230?

1, 2, 5, 10, 823, 1646, 4115, 8230

There is a total of 8 positive divisors.
The sum of these divisors is 14832.
The arithmetic mean is 1854.

4 even divisors

2, 10, 1646, 8230

4 odd divisors

1, 5, 823, 4115

How to compute the divisors of 8230?

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 8230 by each of the numbers from 1 to 8230 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)

8230 / 1 = 8230 (the remainder is 0, so 1 is a divisor of 8230)
8230 / 2 = 4115 (the remainder is 0, so 2 is a divisor of 8230)
8230 / 3 = 2743.3333333333 (the remainder is 1, so 3 is not a divisor of 8230)
...
8230 / 8229 = 1.0001215214485 (the remainder is 1, so 8229 is not a divisor of 8230)
8230 / 8230 = 1 (the remainder is 0, so 8230 is a divisor of 8230)

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 8230 (i.e. 90.719347440334). 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 $\frac{N}{d} = D$ )

8230 / 1 = 8230 (the remainder is 0, so 1 and 8230 are divisors of 8230)
8230 / 2 = 4115 (the remainder is 0, so 2 and 4115 are divisors of 8230)
8230 / 3 = 2743.3333333333 (the remainder is 1, so 3 is not a divisor of 8230)
...
8230 / 89 = 92.47191011236 (the remainder is 42, so 89 is not a divisor of 8230)
8230 / 90 = 91.444444444444 (the remainder is 40, so 90 is not a divisor of 8230)