What are the divisors of 504?

1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504

There is a total of 24 positive divisors.
The sum of these divisors is 1560.
The arithmetic mean is 65.

18 even divisors

2, 4, 6, 8, 12, 14, 18, 24, 28, 36, 42, 56, 72, 84, 126, 168, 252, 504

6 odd divisors

1, 3, 7, 9, 21, 63

How to compute the divisors of 504?

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 504 by each of the numbers from 1 to 504 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)

504 / 1 = 504 (the remainder is 0, so 1 is a divisor of 504)
504 / 2 = 252 (the remainder is 0, so 2 is a divisor of 504)
504 / 3 = 168 (the remainder is 0, so 3 is a divisor of 504)
...
504 / 503 = 1.0019880715706 (the remainder is 1, so 503 is not a divisor of 504)
504 / 504 = 1 (the remainder is 0, so 504 is a divisor of 504)

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 504 (i.e. 22.449944320644). 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 $\frac{N}{d} = D$ )

504 / 1 = 504 (the remainder is 0, so 1 and 504 are divisors of 504)
504 / 2 = 252 (the remainder is 0, so 2 and 252 are divisors of 504)
504 / 3 = 168 (the remainder is 0, so 3 and 168 are divisors of 504)
...
504 / 21 = 24 (the remainder is 0, so 21 and 24 are divisors of 504)
504 / 22 = 22.909090909091 (the remainder is 20, so 22 is not a divisor of 504)