What are the divisors of 505?

1, 5, 101, 505

There is a total of 4 positive divisors.
The sum of these divisors is 612.
The arithmetic mean is 153.

4 odd divisors

1, 5, 101, 505

How to compute the divisors of 505?

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 505 by each of the numbers from 1 to 505 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)

505 / 1 = 505 (the remainder is 0, so 1 is a divisor of 505)
505 / 2 = 252.5 (the remainder is 1, so 2 is not a divisor of 505)
505 / 3 = 168.33333333333 (the remainder is 1, so 3 is not a divisor of 505)
...
505 / 504 = 1.0019841269841 (the remainder is 1, so 504 is not a divisor of 505)
505 / 505 = 1 (the remainder is 0, so 505 is a divisor of 505)

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 505 (i.e. 22.472205054244). 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 $\frac{N}{d} = D$ )

505 / 1 = 505 (the remainder is 0, so 1 and 505 are divisors of 505)
505 / 2 = 252.5 (the remainder is 1, so 2 is not a divisor of 505)
505 / 3 = 168.33333333333 (the remainder is 1, so 3 is not a divisor of 505)
...
505 / 21 = 24.047619047619 (the remainder is 1, so 21 is not a divisor of 505)
505 / 22 = 22.954545454545 (the remainder is 21, so 22 is not a divisor of 505)