What are the divisors of 468?

1, 2, 3, 4, 6, 9, 12, 13, 18, 26, 36, 39, 52, 78, 117, 156, 234, 468

There is a total of 18 positive divisors.
The sum of these divisors is 1274.
The arithmetic mean is 70.777777777778.

12 even divisors

2, 4, 6, 12, 18, 26, 36, 52, 78, 156, 234, 468

6 odd divisors

1, 3, 9, 13, 39, 117

How to compute the divisors of 468?

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 468 by each of the numbers from 1 to 468 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)

468 / 1 = 468 (the remainder is 0, so 1 is a divisor of 468)
468 / 2 = 234 (the remainder is 0, so 2 is a divisor of 468)
468 / 3 = 156 (the remainder is 0, so 3 is a divisor of 468)
...
468 / 467 = 1.0021413276231 (the remainder is 1, so 467 is not a divisor of 468)
468 / 468 = 1 (the remainder is 0, so 468 is a divisor of 468)

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 468 (i.e. 21.633307652784). 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 $\frac{N}{d} = D$ )

468 / 1 = 468 (the remainder is 0, so 1 and 468 are divisors of 468)
468 / 2 = 234 (the remainder is 0, so 2 and 234 are divisors of 468)
468 / 3 = 156 (the remainder is 0, so 3 and 156 are divisors of 468)
...
468 / 20 = 23.4 (the remainder is 8, so 20 is not a divisor of 468)
468 / 21 = 22.285714285714 (the remainder is 6, so 21 is not a divisor of 468)