What are the divisors of 854?

1, 2, 7, 14, 61, 122, 427, 854

There is a total of 8 positive divisors.
The sum of these divisors is 1488.
The arithmetic mean is 186.

4 even divisors

2, 14, 122, 854

4 odd divisors

1, 7, 61, 427

How to compute the divisors of 854?

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 854 by each of the numbers from 1 to 854 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)

854 / 1 = 854 (the remainder is 0, so 1 is a divisor of 854)
854 / 2 = 427 (the remainder is 0, so 2 is a divisor of 854)
854 / 3 = 284.66666666667 (the remainder is 2, so 3 is not a divisor of 854)
...
854 / 853 = 1.0011723329426 (the remainder is 1, so 853 is not a divisor of 854)
854 / 854 = 1 (the remainder is 0, so 854 is a divisor of 854)

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 854 (i.e. 29.223278392405). 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 $\frac{N}{d} = D$ )

854 / 1 = 854 (the remainder is 0, so 1 and 854 are divisors of 854)
854 / 2 = 427 (the remainder is 0, so 2 and 427 are divisors of 854)
854 / 3 = 284.66666666667 (the remainder is 2, so 3 is not a divisor of 854)
...
854 / 28 = 30.5 (the remainder is 14, so 28 is not a divisor of 854)
854 / 29 = 29.448275862069 (the remainder is 13, so 29 is not a divisor of 854)