What are the divisors of 8243?

1, 8243

There is a total of 2 positive divisors.
The sum of these divisors is 8244.
The arithmetic mean is 4122.

2 odd divisors

1, 8243

How to compute the divisors of 8243?

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 8243 by each of the numbers from 1 to 8243 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)

8243 / 1 = 8243 (the remainder is 0, so 1 is a divisor of 8243)
8243 / 2 = 4121.5 (the remainder is 1, so 2 is not a divisor of 8243)
8243 / 3 = 2747.6666666667 (the remainder is 2, so 3 is not a divisor of 8243)
...
8243 / 8242 = 1.0001213297743 (the remainder is 1, so 8242 is not a divisor of 8243)
8243 / 8243 = 1 (the remainder is 0, so 8243 is a divisor of 8243)

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 8243 (i.e. 90.790968713854). 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 $\frac{N}{d} = D$ )

8243 / 1 = 8243 (the remainder is 0, so 1 and 8243 are divisors of 8243)
8243 / 2 = 4121.5 (the remainder is 1, so 2 is not a divisor of 8243)
8243 / 3 = 2747.6666666667 (the remainder is 2, so 3 is not a divisor of 8243)
...
8243 / 89 = 92.61797752809 (the remainder is 55, so 89 is not a divisor of 8243)
8243 / 90 = 91.588888888889 (the remainder is 53, so 90 is not a divisor of 8243)