What are the divisors of 8219?

1, 8219

There is a total of 2 positive divisors.
The sum of these divisors is 8220.
The arithmetic mean is 4110.

2 odd divisors

1, 8219

How to compute the divisors of 8219?

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 8219 by each of the numbers from 1 to 8219 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)

8219 / 1 = 8219 (the remainder is 0, so 1 is a divisor of 8219)
8219 / 2 = 4109.5 (the remainder is 1, so 2 is not a divisor of 8219)
8219 / 3 = 2739.6666666667 (the remainder is 2, so 3 is not a divisor of 8219)
...
8219 / 8218 = 1.0001216841081 (the remainder is 1, so 8218 is not a divisor of 8219)
8219 / 8219 = 1 (the remainder is 0, so 8219 is a divisor of 8219)

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 8219 (i.e. 90.658700630441). 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 $\frac{N}{d} = D$ )

8219 / 1 = 8219 (the remainder is 0, so 1 and 8219 are divisors of 8219)
8219 / 2 = 4109.5 (the remainder is 1, so 2 is not a divisor of 8219)
8219 / 3 = 2739.6666666667 (the remainder is 2, so 3 is not a divisor of 8219)
...
8219 / 89 = 92.348314606742 (the remainder is 31, so 89 is not a divisor of 8219)
8219 / 90 = 91.322222222222 (the remainder is 29, so 90 is not a divisor of 8219)