What are the divisors of 825?

1, 3, 5, 11, 15, 25, 33, 55, 75, 165, 275, 825

There is a total of 12 positive divisors.
The sum of these divisors is 1488.
The arithmetic mean is 124.

12 odd divisors

1, 3, 5, 11, 15, 25, 33, 55, 75, 165, 275, 825

How to compute the divisors of 825?

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 825 by each of the numbers from 1 to 825 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)

825 / 1 = 825 (the remainder is 0, so 1 is a divisor of 825)
825 / 2 = 412.5 (the remainder is 1, so 2 is not a divisor of 825)
825 / 3 = 275 (the remainder is 0, so 3 is a divisor of 825)
...
825 / 824 = 1.001213592233 (the remainder is 1, so 824 is not a divisor of 825)
825 / 825 = 1 (the remainder is 0, so 825 is a divisor of 825)

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 825 (i.e. 28.72281323269). 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 $\frac{N}{d} = D$ )

825 / 1 = 825 (the remainder is 0, so 1 and 825 are divisors of 825)
825 / 2 = 412.5 (the remainder is 1, so 2 is not a divisor of 825)
825 / 3 = 275 (the remainder is 0, so 3 and 275 are divisors of 825)
...
825 / 27 = 30.555555555556 (the remainder is 15, so 27 is not a divisor of 825)
825 / 28 = 29.464285714286 (the remainder is 13, so 28 is not a divisor of 825)