What are the divisors of 824?

1, 2, 4, 8, 103, 206, 412, 824

There is a total of 8 positive divisors.
The sum of these divisors is 1560.
The arithmetic mean is 195.

6 even divisors

2, 4, 8, 206, 412, 824

2 odd divisors

1, 103

How to compute the divisors of 824?

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 824 by each of the numbers from 1 to 824 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)

824 / 1 = 824 (the remainder is 0, so 1 is a divisor of 824)
824 / 2 = 412 (the remainder is 0, so 2 is a divisor of 824)
824 / 3 = 274.66666666667 (the remainder is 2, so 3 is not a divisor of 824)
...
824 / 823 = 1.0012150668287 (the remainder is 1, so 823 is not a divisor of 824)
824 / 824 = 1 (the remainder is 0, so 824 is a divisor of 824)

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 824 (i.e. 28.705400188815). 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 $\frac{N}{d} = D$ )

824 / 1 = 824 (the remainder is 0, so 1 and 824 are divisors of 824)
824 / 2 = 412 (the remainder is 0, so 2 and 412 are divisors of 824)
824 / 3 = 274.66666666667 (the remainder is 2, so 3 is not a divisor of 824)
...
824 / 27 = 30.518518518519 (the remainder is 14, so 27 is not a divisor of 824)
824 / 28 = 29.428571428571 (the remainder is 12, so 28 is not a divisor of 824)