What are the divisors of 8239?

1, 7, 11, 77, 107, 749, 1177, 8239

There is a total of 8 positive divisors.
The sum of these divisors is 10368.
The arithmetic mean is 1296.

8 odd divisors

1, 7, 11, 77, 107, 749, 1177, 8239

How to compute the divisors of 8239?

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 8239 by each of the numbers from 1 to 8239 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)

8239 / 1 = 8239 (the remainder is 0, so 1 is a divisor of 8239)
8239 / 2 = 4119.5 (the remainder is 1, so 2 is not a divisor of 8239)
8239 / 3 = 2746.3333333333 (the remainder is 1, so 3 is not a divisor of 8239)
...
8239 / 8238 = 1.0001213886866 (the remainder is 1, so 8238 is not a divisor of 8239)
8239 / 8239 = 1 (the remainder is 0, so 8239 is a divisor of 8239)

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 8239 (i.e. 90.76893741804). 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 $\frac{N}{d} = D$ )

8239 / 1 = 8239 (the remainder is 0, so 1 and 8239 are divisors of 8239)
8239 / 2 = 4119.5 (the remainder is 1, so 2 is not a divisor of 8239)
8239 / 3 = 2746.3333333333 (the remainder is 1, so 3 is not a divisor of 8239)
...
8239 / 89 = 92.573033707865 (the remainder is 51, so 89 is not a divisor of 8239)
8239 / 90 = 91.544444444444 (the remainder is 49, so 90 is not a divisor of 8239)