What are the divisors of 8198?

1, 2, 4099, 8198

There is a total of 4 positive divisors.
The sum of these divisors is 12300.
The arithmetic mean is 3075.

2 even divisors

2, 8198

2 odd divisors

1, 4099

How to compute the divisors of 8198?

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 8198 by each of the numbers from 1 to 8198 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)

8198 / 1 = 8198 (the remainder is 0, so 1 is a divisor of 8198)
8198 / 2 = 4099 (the remainder is 0, so 2 is a divisor of 8198)
8198 / 3 = 2732.6666666667 (the remainder is 2, so 3 is not a divisor of 8198)
...
8198 / 8197 = 1.0001219958521 (the remainder is 1, so 8197 is not a divisor of 8198)
8198 / 8198 = 1 (the remainder is 0, so 8198 is a divisor of 8198)

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 8198 (i.e. 90.542807555322). 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 $\frac{N}{d} = D$ )

8198 / 1 = 8198 (the remainder is 0, so 1 and 8198 are divisors of 8198)
8198 / 2 = 4099 (the remainder is 0, so 2 and 4099 are divisors of 8198)
8198 / 3 = 2732.6666666667 (the remainder is 2, so 3 is not a divisor of 8198)
...
8198 / 89 = 92.112359550562 (the remainder is 10, so 89 is not a divisor of 8198)
8198 / 90 = 91.088888888889 (the remainder is 8, so 90 is not a divisor of 8198)