What are the divisors of 823?

1, 823

There is a total of 2 positive divisors.
The sum of these divisors is 824.
The arithmetic mean is 412.

2 odd divisors

1, 823

How to compute the divisors of 823?

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 823 by each of the numbers from 1 to 823 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)

823 / 1 = 823 (the remainder is 0, so 1 is a divisor of 823)
823 / 2 = 411.5 (the remainder is 1, so 2 is not a divisor of 823)
823 / 3 = 274.33333333333 (the remainder is 1, so 3 is not a divisor of 823)
...
823 / 822 = 1.0012165450122 (the remainder is 1, so 822 is not a divisor of 823)
823 / 823 = 1 (the remainder is 0, so 823 is a divisor of 823)

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 823 (i.e. 28.687976575562). 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 $\frac{N}{d} = D$ )

823 / 1 = 823 (the remainder is 0, so 1 and 823 are divisors of 823)
823 / 2 = 411.5 (the remainder is 1, so 2 is not a divisor of 823)
823 / 3 = 274.33333333333 (the remainder is 1, so 3 is not a divisor of 823)
...
823 / 27 = 30.481481481481 (the remainder is 13, so 27 is not a divisor of 823)
823 / 28 = 29.392857142857 (the remainder is 11, so 28 is not a divisor of 823)