What are the divisors of 8069?

1, 8069

There is a total of 2 positive divisors.
The sum of these divisors is 8070.
The arithmetic mean is 4035.

2 odd divisors

1, 8069

How to compute the divisors of 8069?

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 8069 by each of the numbers from 1 to 8069 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)

8069 / 1 = 8069 (the remainder is 0, so 1 is a divisor of 8069)
8069 / 2 = 4034.5 (the remainder is 1, so 2 is not a divisor of 8069)
8069 / 3 = 2689.6666666667 (the remainder is 2, so 3 is not a divisor of 8069)
...
8069 / 8068 = 1.0001239464551 (the remainder is 1, so 8068 is not a divisor of 8069)
8069 / 8069 = 1 (the remainder is 0, so 8069 is a divisor of 8069)

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 8069 (i.e. 89.82761268118). 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 $\frac{N}{d} = D$ )

8069 / 1 = 8069 (the remainder is 0, so 1 and 8069 are divisors of 8069)
8069 / 2 = 4034.5 (the remainder is 1, so 2 is not a divisor of 8069)
8069 / 3 = 2689.6666666667 (the remainder is 2, so 3 is not a divisor of 8069)
...
8069 / 88 = 91.693181818182 (the remainder is 61, so 88 is not a divisor of 8069)
8069 / 89 = 90.662921348315 (the remainder is 59, so 89 is not a divisor of 8069)