What are the divisors of 999?

1, 3, 9, 27, 37, 111, 333, 999

There is a total of 8 positive divisors.
The sum of these divisors is 1520.
The arithmetic mean is 190.

8 odd divisors

1, 3, 9, 27, 37, 111, 333, 999

How to compute the divisors of 999?

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 999 by each of the numbers from 1 to 999 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)

999 / 1 = 999 (the remainder is 0, so 1 is a divisor of 999)
999 / 2 = 499.5 (the remainder is 1, so 2 is not a divisor of 999)
999 / 3 = 333 (the remainder is 0, so 3 is a divisor of 999)
...
999 / 998 = 1.001002004008 (the remainder is 1, so 998 is not a divisor of 999)
999 / 999 = 1 (the remainder is 0, so 999 is a divisor of 999)

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 999 (i.e. 31.606961258558). 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 $\frac{N}{d} = D$ )

999 / 1 = 999 (the remainder is 0, so 1 and 999 are divisors of 999)
999 / 2 = 499.5 (the remainder is 1, so 2 is not a divisor of 999)
999 / 3 = 333 (the remainder is 0, so 3 and 333 are divisors of 999)
...
999 / 30 = 33.3 (the remainder is 9, so 30 is not a divisor of 999)
999 / 31 = 32.225806451613 (the remainder is 7, so 31 is not a divisor of 999)