What are the divisors of 459?

1, 3, 9, 17, 27, 51, 153, 459

There is a total of 8 positive divisors.
The sum of these divisors is 720.
The arithmetic mean is 90.

8 odd divisors

1, 3, 9, 17, 27, 51, 153, 459

How to compute the divisors of 459?

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 459 by each of the numbers from 1 to 459 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)

459 / 1 = 459 (the remainder is 0, so 1 is a divisor of 459)
459 / 2 = 229.5 (the remainder is 1, so 2 is not a divisor of 459)
459 / 3 = 153 (the remainder is 0, so 3 is a divisor of 459)
...
459 / 458 = 1.0021834061135 (the remainder is 1, so 458 is not a divisor of 459)
459 / 459 = 1 (the remainder is 0, so 459 is a divisor of 459)

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 459 (i.e. 21.424285285629). 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 $\frac{N}{d} = D$ )

459 / 1 = 459 (the remainder is 0, so 1 and 459 are divisors of 459)
459 / 2 = 229.5 (the remainder is 1, so 2 is not a divisor of 459)
459 / 3 = 153 (the remainder is 0, so 3 and 153 are divisors of 459)
...
459 / 20 = 22.95 (the remainder is 19, so 20 is not a divisor of 459)
459 / 21 = 21.857142857143 (the remainder is 18, so 21 is not a divisor of 459)