What are the divisors of 8207?

1, 29, 283, 8207

There is a total of 4 positive divisors.
The sum of these divisors is 8520.
The arithmetic mean is 2130.

4 odd divisors

1, 29, 283, 8207

How to compute the divisors of 8207?

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 8207 by each of the numbers from 1 to 8207 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)

8207 / 1 = 8207 (the remainder is 0, so 1 is a divisor of 8207)
8207 / 2 = 4103.5 (the remainder is 1, so 2 is not a divisor of 8207)
8207 / 3 = 2735.6666666667 (the remainder is 2, so 3 is not a divisor of 8207)
...
8207 / 8206 = 1.0001218620522 (the remainder is 1, so 8206 is not a divisor of 8207)
8207 / 8207 = 1 (the remainder is 0, so 8207 is a divisor of 8207)

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 8207 (i.e. 90.592494170323). 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 $\frac{N}{d} = D$ )

8207 / 1 = 8207 (the remainder is 0, so 1 and 8207 are divisors of 8207)
8207 / 2 = 4103.5 (the remainder is 1, so 2 is not a divisor of 8207)
8207 / 3 = 2735.6666666667 (the remainder is 2, so 3 is not a divisor of 8207)
...
8207 / 89 = 92.213483146067 (the remainder is 19, so 89 is not a divisor of 8207)
8207 / 90 = 91.188888888889 (the remainder is 17, so 90 is not a divisor of 8207)