What are the divisors of 8123?

1, 8123

There is a total of 2 positive divisors.
The sum of these divisors is 8124.
The arithmetic mean is 4062.

2 odd divisors

1, 8123

How to compute the divisors of 8123?

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 8123 by each of the numbers from 1 to 8123 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)

8123 / 1 = 8123 (the remainder is 0, so 1 is a divisor of 8123)
8123 / 2 = 4061.5 (the remainder is 1, so 2 is not a divisor of 8123)
8123 / 3 = 2707.6666666667 (the remainder is 2, so 3 is not a divisor of 8123)
...
8123 / 8122 = 1.0001231223836 (the remainder is 1, so 8122 is not a divisor of 8123)
8123 / 8123 = 1 (the remainder is 0, so 8123 is a divisor of 8123)

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 8123 (i.e. 90.127687199883). 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 $\frac{N}{d} = D$ )

8123 / 1 = 8123 (the remainder is 0, so 1 and 8123 are divisors of 8123)
8123 / 2 = 4061.5 (the remainder is 1, so 2 is not a divisor of 8123)
8123 / 3 = 2707.6666666667 (the remainder is 2, so 3 is not a divisor of 8123)
...
8123 / 89 = 91.269662921348 (the remainder is 24, so 89 is not a divisor of 8123)
8123 / 90 = 90.255555555556 (the remainder is 23, so 90 is not a divisor of 8123)