What are the divisors of 465?

1, 3, 5, 15, 31, 93, 155, 465

There is a total of 8 positive divisors.
The sum of these divisors is 768.
The arithmetic mean is 96.

8 odd divisors

1, 3, 5, 15, 31, 93, 155, 465

How to compute the divisors of 465?

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 465 by each of the numbers from 1 to 465 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)

465 / 1 = 465 (the remainder is 0, so 1 is a divisor of 465)
465 / 2 = 232.5 (the remainder is 1, so 2 is not a divisor of 465)
465 / 3 = 155 (the remainder is 0, so 3 is a divisor of 465)
...
465 / 464 = 1.0021551724138 (the remainder is 1, so 464 is not a divisor of 465)
465 / 465 = 1 (the remainder is 0, so 465 is a divisor of 465)

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 465 (i.e. 21.563858652848). 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 $\frac{N}{d} = D$ )

465 / 1 = 465 (the remainder is 0, so 1 and 465 are divisors of 465)
465 / 2 = 232.5 (the remainder is 1, so 2 is not a divisor of 465)
465 / 3 = 155 (the remainder is 0, so 3 and 155 are divisors of 465)
...
465 / 20 = 23.25 (the remainder is 5, so 20 is not a divisor of 465)
465 / 21 = 22.142857142857 (the remainder is 3, so 21 is not a divisor of 465)