What are the divisors of 815?

1, 5, 163, 815

There is a total of 4 positive divisors.
The sum of these divisors is 984.
The arithmetic mean is 246.

4 odd divisors

1, 5, 163, 815

How to compute the divisors of 815?

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 815 by each of the numbers from 1 to 815 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)

815 / 1 = 815 (the remainder is 0, so 1 is a divisor of 815)
815 / 2 = 407.5 (the remainder is 1, so 2 is not a divisor of 815)
815 / 3 = 271.66666666667 (the remainder is 2, so 3 is not a divisor of 815)
...
815 / 814 = 1.0012285012285 (the remainder is 1, so 814 is not a divisor of 815)
815 / 815 = 1 (the remainder is 0, so 815 is a divisor of 815)

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 815 (i.e. 28.54820484724). 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 $\frac{N}{d} = D$ )

815 / 1 = 815 (the remainder is 0, so 1 and 815 are divisors of 815)
815 / 2 = 407.5 (the remainder is 1, so 2 is not a divisor of 815)
815 / 3 = 271.66666666667 (the remainder is 2, so 3 is not a divisor of 815)
...
815 / 27 = 30.185185185185 (the remainder is 5, so 27 is not a divisor of 815)
815 / 28 = 29.107142857143 (the remainder is 3, so 28 is not a divisor of 815)