What are the divisors of 826?

1, 2, 7, 14, 59, 118, 413, 826

There is a total of 8 positive divisors.
The sum of these divisors is 1440.
The arithmetic mean is 180.

4 even divisors

2, 14, 118, 826

4 odd divisors

1, 7, 59, 413

How to compute the divisors of 826?

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 826 by each of the numbers from 1 to 826 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)

826 / 1 = 826 (the remainder is 0, so 1 is a divisor of 826)
826 / 2 = 413 (the remainder is 0, so 2 is a divisor of 826)
826 / 3 = 275.33333333333 (the remainder is 1, so 3 is not a divisor of 826)
...
826 / 825 = 1.0012121212121 (the remainder is 1, so 825 is not a divisor of 826)
826 / 826 = 1 (the remainder is 0, so 826 is a divisor of 826)

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 826 (i.e. 28.7402157264). 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 $\frac{N}{d} = D$ )

826 / 1 = 826 (the remainder is 0, so 1 and 826 are divisors of 826)
826 / 2 = 413 (the remainder is 0, so 2 and 413 are divisors of 826)
826 / 3 = 275.33333333333 (the remainder is 1, so 3 is not a divisor of 826)
...
826 / 27 = 30.592592592593 (the remainder is 16, so 27 is not a divisor of 826)
826 / 28 = 29.5 (the remainder is 14, so 28 is not a divisor of 826)