What are the divisors of 8122?

1, 2, 31, 62, 131, 262, 4061, 8122

There is a total of 8 positive divisors.
The sum of these divisors is 12672.
The arithmetic mean is 1584.

4 even divisors

2, 62, 262, 8122

4 odd divisors

1, 31, 131, 4061

How to compute the divisors of 8122?

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 8122 by each of the numbers from 1 to 8122 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)

8122 / 1 = 8122 (the remainder is 0, so 1 is a divisor of 8122)
8122 / 2 = 4061 (the remainder is 0, so 2 is a divisor of 8122)
8122 / 3 = 2707.3333333333 (the remainder is 1, so 3 is not a divisor of 8122)
...
8122 / 8121 = 1.0001231375446 (the remainder is 1, so 8121 is not a divisor of 8122)
8122 / 8122 = 1 (the remainder is 0, so 8122 is a divisor of 8122)

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 8122 (i.e. 90.122139344336). 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 $\frac{N}{d} = D$ )

8122 / 1 = 8122 (the remainder is 0, so 1 and 8122 are divisors of 8122)
8122 / 2 = 4061 (the remainder is 0, so 2 and 4061 are divisors of 8122)
8122 / 3 = 2707.3333333333 (the remainder is 1, so 3 is not a divisor of 8122)
...
8122 / 89 = 91.258426966292 (the remainder is 23, so 89 is not a divisor of 8122)
8122 / 90 = 90.244444444444 (the remainder is 22, so 90 is not a divisor of 8122)