What are the divisors of 8237?

1, 8237

There is a total of 2 positive divisors.
The sum of these divisors is 8238.
The arithmetic mean is 4119.

2 odd divisors

1, 8237

How to compute the divisors of 8237?

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 8237 by each of the numbers from 1 to 8237 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)

8237 / 1 = 8237 (the remainder is 0, so 1 is a divisor of 8237)
8237 / 2 = 4118.5 (the remainder is 1, so 2 is not a divisor of 8237)
8237 / 3 = 2745.6666666667 (the remainder is 2, so 3 is not a divisor of 8237)
...
8237 / 8236 = 1.0001214181642 (the remainder is 1, so 8236 is not a divisor of 8237)
8237 / 8237 = 1 (the remainder is 0, so 8237 is a divisor of 8237)

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 8237 (i.e. 90.757919764613). 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 $\frac{N}{d} = D$ )

8237 / 1 = 8237 (the remainder is 0, so 1 and 8237 are divisors of 8237)
8237 / 2 = 4118.5 (the remainder is 1, so 2 is not a divisor of 8237)
8237 / 3 = 2745.6666666667 (the remainder is 2, so 3 is not a divisor of 8237)
...
8237 / 89 = 92.550561797753 (the remainder is 49, so 89 is not a divisor of 8237)
8237 / 90 = 91.522222222222 (the remainder is 47, so 90 is not a divisor of 8237)