What are the divisors of 8269?

1, 8269

There is a total of 2 positive divisors.
The sum of these divisors is 8270.
The arithmetic mean is 4135.

2 odd divisors

1, 8269

How to compute the divisors of 8269?

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 8269 by each of the numbers from 1 to 8269 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)

8269 / 1 = 8269 (the remainder is 0, so 1 is a divisor of 8269)
8269 / 2 = 4134.5 (the remainder is 1, so 2 is not a divisor of 8269)
8269 / 3 = 2756.3333333333 (the remainder is 1, so 3 is not a divisor of 8269)
...
8269 / 8268 = 1.0001209482342 (the remainder is 1, so 8268 is not a divisor of 8269)
8269 / 8269 = 1 (the remainder is 0, so 8269 is a divisor of 8269)

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 8269 (i.e. 90.934042030474). 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 $\frac{N}{d} = D$ )

8269 / 1 = 8269 (the remainder is 0, so 1 and 8269 are divisors of 8269)
8269 / 2 = 4134.5 (the remainder is 1, so 2 is not a divisor of 8269)
8269 / 3 = 2756.3333333333 (the remainder is 1, so 3 is not a divisor of 8269)
...
8269 / 89 = 92.910112359551 (the remainder is 81, so 89 is not a divisor of 8269)
8269 / 90 = 91.877777777778 (the remainder is 79, so 90 is not a divisor of 8269)