What are the divisors of 803?

1, 11, 73, 803

There is a total of 4 positive divisors.
The sum of these divisors is 888.
The arithmetic mean is 222.

4 odd divisors

1, 11, 73, 803

How to compute the divisors of 803?

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 803 by each of the numbers from 1 to 803 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)

803 / 1 = 803 (the remainder is 0, so 1 is a divisor of 803)
803 / 2 = 401.5 (the remainder is 1, so 2 is not a divisor of 803)
803 / 3 = 267.66666666667 (the remainder is 2, so 3 is not a divisor of 803)
...
803 / 802 = 1.001246882793 (the remainder is 1, so 802 is not a divisor of 803)
803 / 803 = 1 (the remainder is 0, so 803 is a divisor of 803)

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 803 (i.e. 28.33725463061). 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 $\frac{N}{d} = D$ )

803 / 1 = 803 (the remainder is 0, so 1 and 803 are divisors of 803)
803 / 2 = 401.5 (the remainder is 1, so 2 is not a divisor of 803)
803 / 3 = 267.66666666667 (the remainder is 2, so 3 is not a divisor of 803)
...
803 / 27 = 29.740740740741 (the remainder is 20, so 27 is not a divisor of 803)
803 / 28 = 28.678571428571 (the remainder is 19, so 28 is not a divisor of 803)