What are the divisors of 8214?

1, 2, 3, 6, 37, 74, 111, 222, 1369, 2738, 4107, 8214

There is a total of 12 positive divisors.
The sum of these divisors is 16884.
The arithmetic mean is 1407.

6 even divisors

2, 6, 74, 222, 2738, 8214

6 odd divisors

1, 3, 37, 111, 1369, 4107

How to compute the divisors of 8214?

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 8214 by each of the numbers from 1 to 8214 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)

8214 / 1 = 8214 (the remainder is 0, so 1 is a divisor of 8214)
8214 / 2 = 4107 (the remainder is 0, so 2 is a divisor of 8214)
8214 / 3 = 2738 (the remainder is 0, so 3 is a divisor of 8214)
...
8214 / 8213 = 1.0001217581882 (the remainder is 1, so 8213 is not a divisor of 8214)
8214 / 8214 = 1 (the remainder is 0, so 8214 is a divisor of 8214)

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 8214 (i.e. 90.631120482978). 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 $\frac{N}{d} = D$ )

8214 / 1 = 8214 (the remainder is 0, so 1 and 8214 are divisors of 8214)
8214 / 2 = 4107 (the remainder is 0, so 2 and 4107 are divisors of 8214)
8214 / 3 = 2738 (the remainder is 0, so 3 and 2738 are divisors of 8214)
...
8214 / 89 = 92.292134831461 (the remainder is 26, so 89 is not a divisor of 8214)
8214 / 90 = 91.266666666667 (the remainder is 24, so 90 is not a divisor of 8214)