What are the divisors of 8222?

1, 2, 4111, 8222

There is a total of 4 positive divisors.
The sum of these divisors is 12336.
The arithmetic mean is 3084.

2 even divisors

2, 8222

2 odd divisors

1, 4111

How to compute the divisors of 8222?

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 8222 by each of the numbers from 1 to 8222 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)

8222 / 1 = 8222 (the remainder is 0, so 1 is a divisor of 8222)
8222 / 2 = 4111 (the remainder is 0, so 2 is a divisor of 8222)
8222 / 3 = 2740.6666666667 (the remainder is 2, so 3 is not a divisor of 8222)
...
8222 / 8221 = 1.0001216397032 (the remainder is 1, so 8221 is not a divisor of 8222)
8222 / 8222 = 1 (the remainder is 0, so 8222 is a divisor of 8222)

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 8222 (i.e. 90.675244692253). 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 $\frac{N}{d} = D$ )

8222 / 1 = 8222 (the remainder is 0, so 1 and 8222 are divisors of 8222)
8222 / 2 = 4111 (the remainder is 0, so 2 and 4111 are divisors of 8222)
8222 / 3 = 2740.6666666667 (the remainder is 2, so 3 is not a divisor of 8222)
...
8222 / 89 = 92.38202247191 (the remainder is 34, so 89 is not a divisor of 8222)
8222 / 90 = 91.355555555556 (the remainder is 32, so 90 is not a divisor of 8222)