What are the divisors of 495?

1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495

There is a total of 12 positive divisors.
The sum of these divisors is 936.
The arithmetic mean is 78.

12 odd divisors

1, 3, 5, 9, 11, 15, 33, 45, 55, 99, 165, 495

How to compute the divisors of 495?

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 495 by each of the numbers from 1 to 495 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)

495 / 1 = 495 (the remainder is 0, so 1 is a divisor of 495)
495 / 2 = 247.5 (the remainder is 1, so 2 is not a divisor of 495)
495 / 3 = 165 (the remainder is 0, so 3 is a divisor of 495)
...
495 / 494 = 1.002024291498 (the remainder is 1, so 494 is not a divisor of 495)
495 / 495 = 1 (the remainder is 0, so 495 is a divisor of 495)

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 495 (i.e. 22.248595461287). 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 $\frac{N}{d} = D$ )

495 / 1 = 495 (the remainder is 0, so 1 and 495 are divisors of 495)
495 / 2 = 247.5 (the remainder is 1, so 2 is not a divisor of 495)
495 / 3 = 165 (the remainder is 0, so 3 and 165 are divisors of 495)
...
495 / 21 = 23.571428571429 (the remainder is 12, so 21 is not a divisor of 495)
495 / 22 = 22.5 (the remainder is 11, so 22 is not a divisor of 495)