What are the divisors of 829?

1, 829

There is a total of 2 positive divisors.
The sum of these divisors is 830.
The arithmetic mean is 415.

2 odd divisors

1, 829

How to compute the divisors of 829?

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 829 by each of the numbers from 1 to 829 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)

829 / 1 = 829 (the remainder is 0, so 1 is a divisor of 829)
829 / 2 = 414.5 (the remainder is 1, so 2 is not a divisor of 829)
829 / 3 = 276.33333333333 (the remainder is 1, so 3 is not a divisor of 829)
...
829 / 828 = 1.0012077294686 (the remainder is 1, so 828 is not a divisor of 829)
829 / 829 = 1 (the remainder is 0, so 829 is a divisor of 829)

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 829 (i.e. 28.792360097776). 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 $\frac{N}{d} = D$ )

829 / 1 = 829 (the remainder is 0, so 1 and 829 are divisors of 829)
829 / 2 = 414.5 (the remainder is 1, so 2 is not a divisor of 829)
829 / 3 = 276.33333333333 (the remainder is 1, so 3 is not a divisor of 829)
...
829 / 27 = 30.703703703704 (the remainder is 19, so 27 is not a divisor of 829)
829 / 28 = 29.607142857143 (the remainder is 17, so 28 is not a divisor of 829)