What are the divisors of 503?

1, 503

There is a total of 2 positive divisors.
The sum of these divisors is 504.
The arithmetic mean is 252.

2 odd divisors

1, 503

How to compute the divisors of 503?

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 503 by each of the numbers from 1 to 503 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)

503 / 1 = 503 (the remainder is 0, so 1 is a divisor of 503)
503 / 2 = 251.5 (the remainder is 1, so 2 is not a divisor of 503)
503 / 3 = 167.66666666667 (the remainder is 2, so 3 is not a divisor of 503)
...
503 / 502 = 1.0019920318725 (the remainder is 1, so 502 is not a divisor of 503)
503 / 503 = 1 (the remainder is 0, so 503 is a divisor of 503)

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 503 (i.e. 22.427661492006). 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 $\frac{N}{d} = D$ )

503 / 1 = 503 (the remainder is 0, so 1 and 503 are divisors of 503)
503 / 2 = 251.5 (the remainder is 1, so 2 is not a divisor of 503)
503 / 3 = 167.66666666667 (the remainder is 2, so 3 is not a divisor of 503)
...
503 / 21 = 23.952380952381 (the remainder is 20, so 21 is not a divisor of 503)
503 / 22 = 22.863636363636 (the remainder is 19, so 22 is not a divisor of 503)