What are the divisors of 1329?

1, 3, 443, 1329

There is a total of 4 positive divisors.
The sum of these divisors is 1776.
The arithmetic mean is 444.

4 odd divisors

1, 3, 443, 1329

How to compute the divisors of 1329?

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 1329 by each of the numbers from 1 to 1329 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)

1329 / 1 = 1329 (the remainder is 0, so 1 is a divisor of 1329)
1329 / 2 = 664.5 (the remainder is 1, so 2 is not a divisor of 1329)
1329 / 3 = 443 (the remainder is 0, so 3 is a divisor of 1329)
...
1329 / 1328 = 1.0007530120482 (the remainder is 1, so 1328 is not a divisor of 1329)
1329 / 1329 = 1 (the remainder is 0, so 1329 is a divisor of 1329)

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 1329 (i.e. 36.455452267116). 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 $\frac{N}{d} = D$ )

1329 / 1 = 1329 (the remainder is 0, so 1 and 1329 are divisors of 1329)
1329 / 2 = 664.5 (the remainder is 1, so 2 is not a divisor of 1329)
1329 / 3 = 443 (the remainder is 0, so 3 and 443 are divisors of 1329)
...
1329 / 35 = 37.971428571429 (the remainder is 34, so 35 is not a divisor of 1329)
1329 / 36 = 36.916666666667 (the remainder is 33, so 36 is not a divisor of 1329)