What are the divisors of 806?

1, 2, 13, 26, 31, 62, 403, 806

There is a total of 8 positive divisors.
The sum of these divisors is 1344.
The arithmetic mean is 168.

4 even divisors

2, 26, 62, 806

4 odd divisors

1, 13, 31, 403

How to compute the divisors of 806?

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 806 by each of the numbers from 1 to 806 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)

806 / 1 = 806 (the remainder is 0, so 1 is a divisor of 806)
806 / 2 = 403 (the remainder is 0, so 2 is a divisor of 806)
806 / 3 = 268.66666666667 (the remainder is 2, so 3 is not a divisor of 806)
...
806 / 805 = 1.0012422360248 (the remainder is 1, so 805 is not a divisor of 806)
806 / 806 = 1 (the remainder is 0, so 806 is a divisor of 806)

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 806 (i.e. 28.390139133157). 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 $\frac{N}{d} = D$ )

806 / 1 = 806 (the remainder is 0, so 1 and 806 are divisors of 806)
806 / 2 = 403 (the remainder is 0, so 2 and 403 are divisors of 806)
806 / 3 = 268.66666666667 (the remainder is 2, so 3 is not a divisor of 806)
...
806 / 27 = 29.851851851852 (the remainder is 23, so 27 is not a divisor of 806)
806 / 28 = 28.785714285714 (the remainder is 22, so 28 is not a divisor of 806)