What are the divisors of 460?

1, 2, 4, 5, 10, 20, 23, 46, 92, 115, 230, 460

There is a total of 12 positive divisors.
The sum of these divisors is 1008.
The arithmetic mean is 84.

8 even divisors

2, 4, 10, 20, 46, 92, 230, 460

4 odd divisors

1, 5, 23, 115

How to compute the divisors of 460?

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 460 by each of the numbers from 1 to 460 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)

460 / 1 = 460 (the remainder is 0, so 1 is a divisor of 460)
460 / 2 = 230 (the remainder is 0, so 2 is a divisor of 460)
460 / 3 = 153.33333333333 (the remainder is 1, so 3 is not a divisor of 460)
...
460 / 459 = 1.0021786492375 (the remainder is 1, so 459 is not a divisor of 460)
460 / 460 = 1 (the remainder is 0, so 460 is a divisor of 460)

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 460 (i.e. 21.447610589527). 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 $\frac{N}{d} = D$ )

460 / 1 = 460 (the remainder is 0, so 1 and 460 are divisors of 460)
460 / 2 = 230 (the remainder is 0, so 2 and 230 are divisors of 460)
460 / 3 = 153.33333333333 (the remainder is 1, so 3 is not a divisor of 460)
...
460 / 20 = 23 (the remainder is 0, so 20 and 23 are divisors of 460)
460 / 21 = 21.904761904762 (the remainder is 19, so 21 is not a divisor of 460)