What are the divisors of 8164?

1, 2, 4, 13, 26, 52, 157, 314, 628, 2041, 4082, 8164

There is a total of 12 positive divisors.
The sum of these divisors is 15484.
The arithmetic mean is 1290.3333333333.

8 even divisors

2, 4, 26, 52, 314, 628, 4082, 8164

4 odd divisors

1, 13, 157, 2041

How to compute the divisors of 8164?

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 8164 by each of the numbers from 1 to 8164 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)

8164 / 1 = 8164 (the remainder is 0, so 1 is a divisor of 8164)
8164 / 2 = 4082 (the remainder is 0, so 2 is a divisor of 8164)
8164 / 3 = 2721.3333333333 (the remainder is 1, so 3 is not a divisor of 8164)
...
8164 / 8163 = 1.0001225039814 (the remainder is 1, so 8163 is not a divisor of 8164)
8164 / 8164 = 1 (the remainder is 0, so 8164 is a divisor of 8164)

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 8164 (i.e. 90.354855984612). 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 $\frac{N}{d} = D$ )

8164 / 1 = 8164 (the remainder is 0, so 1 and 8164 are divisors of 8164)
8164 / 2 = 4082 (the remainder is 0, so 2 and 4082 are divisors of 8164)
8164 / 3 = 2721.3333333333 (the remainder is 1, so 3 is not a divisor of 8164)
...
8164 / 89 = 91.730337078652 (the remainder is 65, so 89 is not a divisor of 8164)
8164 / 90 = 90.711111111111 (the remainder is 64, so 90 is not a divisor of 8164)