What are the divisors of 820?

1, 2, 4, 5, 10, 20, 41, 82, 164, 205, 410, 820

There is a total of 12 positive divisors.
The sum of these divisors is 1764.
The arithmetic mean is 147.

8 even divisors

2, 4, 10, 20, 82, 164, 410, 820

4 odd divisors

1, 5, 41, 205

How to compute the divisors of 820?

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 820 by each of the numbers from 1 to 820 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)

820 / 1 = 820 (the remainder is 0, so 1 is a divisor of 820)
820 / 2 = 410 (the remainder is 0, so 2 is a divisor of 820)
820 / 3 = 273.33333333333 (the remainder is 1, so 3 is not a divisor of 820)
...
820 / 819 = 1.001221001221 (the remainder is 1, so 819 is not a divisor of 820)
820 / 820 = 1 (the remainder is 0, so 820 is a divisor of 820)

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 820 (i.e. 28.635642126553). 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 $\frac{N}{d} = D$ )

820 / 1 = 820 (the remainder is 0, so 1 and 820 are divisors of 820)
820 / 2 = 410 (the remainder is 0, so 2 and 410 are divisors of 820)
820 / 3 = 273.33333333333 (the remainder is 1, so 3 is not a divisor of 820)
...
820 / 27 = 30.37037037037 (the remainder is 10, so 27 is not a divisor of 820)
820 / 28 = 29.285714285714 (the remainder is 8, so 28 is not a divisor of 820)