What are the divisors of 327?

1, 3, 109, 327

There is a total of 4 positive divisors.
The sum of these divisors is 440.
The arithmetic mean is 110.

4 odd divisors

1, 3, 109, 327

How to compute the divisors of 327?

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 327 by each of the numbers from 1 to 327 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)

327 / 1 = 327 (the remainder is 0, so 1 is a divisor of 327)
327 / 2 = 163.5 (the remainder is 1, so 2 is not a divisor of 327)
327 / 3 = 109 (the remainder is 0, so 3 is a divisor of 327)
...
327 / 326 = 1.0030674846626 (the remainder is 1, so 326 is not a divisor of 327)
327 / 327 = 1 (the remainder is 0, so 327 is a divisor of 327)

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 327 (i.e. 18.083141320025). 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 $\frac{N}{d} = D$ )

327 / 1 = 327 (the remainder is 0, so 1 and 327 are divisors of 327)
327 / 2 = 163.5 (the remainder is 1, so 2 is not a divisor of 327)
327 / 3 = 109 (the remainder is 0, so 3 and 109 are divisors of 327)
...
327 / 17 = 19.235294117647 (the remainder is 4, so 17 is not a divisor of 327)
327 / 18 = 18.166666666667 (the remainder is 3, so 18 is not a divisor of 327)