What are the divisors of 216?

1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 36, 54, 72, 108, 216

There is a total of 16 positive divisors.
The sum of these divisors is 600.
The arithmetic mean is 37.5.

12 even divisors

2, 4, 6, 8, 12, 18, 24, 36, 54, 72, 108, 216

4 odd divisors

1, 3, 9, 27

How to compute the divisors of 216?

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 216 by each of the numbers from 1 to 216 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)

216 / 1 = 216 (the remainder is 0, so 1 is a divisor of 216)
216 / 2 = 108 (the remainder is 0, so 2 is a divisor of 216)
216 / 3 = 72 (the remainder is 0, so 3 is a divisor of 216)
...
216 / 215 = 1.0046511627907 (the remainder is 1, so 215 is not a divisor of 216)
216 / 216 = 1 (the remainder is 0, so 216 is a divisor of 216)

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 216 (i.e. 14.696938456699). 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 $\frac{N}{d} = D$ )

216 / 1 = 216 (the remainder is 0, so 1 and 216 are divisors of 216)
216 / 2 = 108 (the remainder is 0, so 2 and 108 are divisors of 216)
216 / 3 = 72 (the remainder is 0, so 3 and 72 are divisors of 216)
...
216 / 13 = 16.615384615385 (the remainder is 8, so 13 is not a divisor of 216)
216 / 14 = 15.428571428571 (the remainder is 6, so 14 is not a divisor of 216)