What are the divisors of 891?

1, 3, 9, 11, 27, 33, 81, 99, 297, 891

There is a total of 10 positive divisors.
The sum of these divisors is 1452.
The arithmetic mean is 145.2.

10 odd divisors

1, 3, 9, 11, 27, 33, 81, 99, 297, 891

How to compute the divisors of 891?

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 891 by each of the numbers from 1 to 891 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)

891 / 1 = 891 (the remainder is 0, so 1 is a divisor of 891)
891 / 2 = 445.5 (the remainder is 1, so 2 is not a divisor of 891)
891 / 3 = 297 (the remainder is 0, so 3 is a divisor of 891)
...
891 / 890 = 1.0011235955056 (the remainder is 1, so 890 is not a divisor of 891)
891 / 891 = 1 (the remainder is 0, so 891 is a divisor of 891)

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 891 (i.e. 29.849623113199). 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 $\frac{N}{d} = D$ )

891 / 1 = 891 (the remainder is 0, so 1 and 891 are divisors of 891)
891 / 2 = 445.5 (the remainder is 1, so 2 is not a divisor of 891)
891 / 3 = 297 (the remainder is 0, so 3 and 297 are divisors of 891)
...
891 / 28 = 31.821428571429 (the remainder is 23, so 28 is not a divisor of 891)
891 / 29 = 30.724137931034 (the remainder is 21, so 29 is not a divisor of 891)