What are the divisors of 861?

1, 3, 7, 21, 41, 123, 287, 861

There is a total of 8 positive divisors.
The sum of these divisors is 1344.
The arithmetic mean is 168.

8 odd divisors

1, 3, 7, 21, 41, 123, 287, 861

How to compute the divisors of 861?

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 861 by each of the numbers from 1 to 861 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)

861 / 1 = 861 (the remainder is 0, so 1 is a divisor of 861)
861 / 2 = 430.5 (the remainder is 1, so 2 is not a divisor of 861)
861 / 3 = 287 (the remainder is 0, so 3 is a divisor of 861)
...
861 / 860 = 1.0011627906977 (the remainder is 1, so 860 is not a divisor of 861)
861 / 861 = 1 (the remainder is 0, so 861 is a divisor of 861)

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 861 (i.e. 29.342801502242). 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 $\frac{N}{d} = D$ )

861 / 1 = 861 (the remainder is 0, so 1 and 861 are divisors of 861)
861 / 2 = 430.5 (the remainder is 1, so 2 is not a divisor of 861)
861 / 3 = 287 (the remainder is 0, so 3 and 287 are divisors of 861)
...
861 / 28 = 30.75 (the remainder is 21, so 28 is not a divisor of 861)
861 / 29 = 29.689655172414 (the remainder is 20, so 29 is not a divisor of 861)