What are the divisors of 8213?

1, 43, 191, 8213

There is a total of 4 positive divisors.
The sum of these divisors is 8448.
The arithmetic mean is 2112.

4 odd divisors

1, 43, 191, 8213

How to compute the divisors of 8213?

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 8213 by each of the numbers from 1 to 8213 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)

8213 / 1 = 8213 (the remainder is 0, so 1 is a divisor of 8213)
8213 / 2 = 4106.5 (the remainder is 1, so 2 is not a divisor of 8213)
8213 / 3 = 2737.6666666667 (the remainder is 2, so 3 is not a divisor of 8213)
...
8213 / 8212 = 1.0001217730151 (the remainder is 1, so 8212 is not a divisor of 8213)
8213 / 8213 = 1 (the remainder is 0, so 8213 is a divisor of 8213)

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 8213 (i.e. 90.625603446267). 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 $\frac{N}{d} = D$ )

8213 / 1 = 8213 (the remainder is 0, so 1 and 8213 are divisors of 8213)
8213 / 2 = 4106.5 (the remainder is 1, so 2 is not a divisor of 8213)
8213 / 3 = 2737.6666666667 (the remainder is 2, so 3 is not a divisor of 8213)
...
8213 / 89 = 92.280898876404 (the remainder is 25, so 89 is not a divisor of 8213)
8213 / 90 = 91.255555555556 (the remainder is 23, so 90 is not a divisor of 8213)