What are the divisors of 8189?

1, 19, 431, 8189

There is a total of 4 positive divisors.
The sum of these divisors is 8640.
The arithmetic mean is 2160.

4 odd divisors

1, 19, 431, 8189

How to compute the divisors of 8189?

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 8189 by each of the numbers from 1 to 8189 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)

8189 / 1 = 8189 (the remainder is 0, so 1 is a divisor of 8189)
8189 / 2 = 4094.5 (the remainder is 1, so 2 is not a divisor of 8189)
8189 / 3 = 2729.6666666667 (the remainder is 2, so 3 is not a divisor of 8189)
...
8189 / 8188 = 1.0001221299463 (the remainder is 1, so 8188 is not a divisor of 8189)
8189 / 8189 = 1 (the remainder is 0, so 8189 is a divisor of 8189)

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 8189 (i.e. 90.49309365913). 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 $\frac{N}{d} = D$ )

8189 / 1 = 8189 (the remainder is 0, so 1 and 8189 are divisors of 8189)
8189 / 2 = 4094.5 (the remainder is 1, so 2 is not a divisor of 8189)
8189 / 3 = 2729.6666666667 (the remainder is 2, so 3 is not a divisor of 8189)
...
8189 / 89 = 92.011235955056 (the remainder is 1, so 89 is not a divisor of 8189)
8189 / 90 = 90.988888888889 (the remainder is 89, so 90 is not a divisor of 8189)