What are the divisors of 857?

1, 857

There is a total of 2 positive divisors.
The sum of these divisors is 858.
The arithmetic mean is 429.

2 odd divisors

1, 857

How to compute the divisors of 857?

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 857 by each of the numbers from 1 to 857 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)

857 / 1 = 857 (the remainder is 0, so 1 is a divisor of 857)
857 / 2 = 428.5 (the remainder is 1, so 2 is not a divisor of 857)
857 / 3 = 285.66666666667 (the remainder is 2, so 3 is not a divisor of 857)
...
857 / 856 = 1.0011682242991 (the remainder is 1, so 856 is not a divisor of 857)
857 / 857 = 1 (the remainder is 0, so 857 is a divisor of 857)

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 857 (i.e. 29.274562336609). 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 $\frac{N}{d} = D$ )

857 / 1 = 857 (the remainder is 0, so 1 and 857 are divisors of 857)
857 / 2 = 428.5 (the remainder is 1, so 2 is not a divisor of 857)
857 / 3 = 285.66666666667 (the remainder is 2, so 3 is not a divisor of 857)
...
857 / 28 = 30.607142857143 (the remainder is 17, so 28 is not a divisor of 857)
857 / 29 = 29.551724137931 (the remainder is 16, so 29 is not a divisor of 857)