What are the divisors of 814?

1, 2, 11, 22, 37, 74, 407, 814

There is a total of 8 positive divisors.
The sum of these divisors is 1368.
The arithmetic mean is 171.

4 even divisors

2, 22, 74, 814

4 odd divisors

1, 11, 37, 407

How to compute the divisors of 814?

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 814 by each of the numbers from 1 to 814 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)

814 / 1 = 814 (the remainder is 0, so 1 is a divisor of 814)
814 / 2 = 407 (the remainder is 0, so 2 is a divisor of 814)
814 / 3 = 271.33333333333 (the remainder is 1, so 3 is not a divisor of 814)
...
814 / 813 = 1.0012300123001 (the remainder is 1, so 813 is not a divisor of 814)
814 / 814 = 1 (the remainder is 0, so 814 is a divisor of 814)

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 814 (i.e. 28.530685235374). 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 $\frac{N}{d} = D$ )

814 / 1 = 814 (the remainder is 0, so 1 and 814 are divisors of 814)
814 / 2 = 407 (the remainder is 0, so 2 and 407 are divisors of 814)
814 / 3 = 271.33333333333 (the remainder is 1, so 3 is not a divisor of 814)
...
814 / 27 = 30.148148148148 (the remainder is 4, so 27 is not a divisor of 814)
814 / 28 = 29.071428571429 (the remainder is 2, so 28 is not a divisor of 814)