What are the divisors of 8197?

1, 7, 1171, 8197

There is a total of 4 positive divisors.
The sum of these divisors is 9376.
The arithmetic mean is 2344.

4 odd divisors

1, 7, 1171, 8197

How to compute the divisors of 8197?

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 8197 by each of the numbers from 1 to 8197 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)

8197 / 1 = 8197 (the remainder is 0, so 1 is a divisor of 8197)
8197 / 2 = 4098.5 (the remainder is 1, so 2 is not a divisor of 8197)
8197 / 3 = 2732.3333333333 (the remainder is 1, so 3 is not a divisor of 8197)
...
8197 / 8196 = 1.0001220107369 (the remainder is 1, so 8196 is not a divisor of 8197)
8197 / 8197 = 1 (the remainder is 0, so 8197 is a divisor of 8197)

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 8197 (i.e. 90.537285137119). 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 $\frac{N}{d} = D$ )

8197 / 1 = 8197 (the remainder is 0, so 1 and 8197 are divisors of 8197)
8197 / 2 = 4098.5 (the remainder is 1, so 2 is not a divisor of 8197)
8197 / 3 = 2732.3333333333 (the remainder is 1, so 3 is not a divisor of 8197)
...
8197 / 89 = 92.101123595506 (the remainder is 9, so 89 is not a divisor of 8197)
8197 / 90 = 91.077777777778 (the remainder is 7, so 90 is not a divisor of 8197)