What are the divisors of 501?

1, 3, 167, 501

There is a total of 4 positive divisors.
The sum of these divisors is 672.
The arithmetic mean is 168.

4 odd divisors

1, 3, 167, 501

How to compute the divisors of 501?

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 501 by each of the numbers from 1 to 501 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)

501 / 1 = 501 (the remainder is 0, so 1 is a divisor of 501)
501 / 2 = 250.5 (the remainder is 1, so 2 is not a divisor of 501)
501 / 3 = 167 (the remainder is 0, so 3 is a divisor of 501)
...
501 / 500 = 1.002 (the remainder is 1, so 500 is not a divisor of 501)
501 / 501 = 1 (the remainder is 0, so 501 is a divisor of 501)

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 501 (i.e. 22.383029285599). 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 $\frac{N}{d} = D$ )

501 / 1 = 501 (the remainder is 0, so 1 and 501 are divisors of 501)
501 / 2 = 250.5 (the remainder is 1, so 2 is not a divisor of 501)
501 / 3 = 167 (the remainder is 0, so 3 and 167 are divisors of 501)
...
501 / 21 = 23.857142857143 (the remainder is 18, so 21 is not a divisor of 501)
501 / 22 = 22.772727272727 (the remainder is 17, so 22 is not a divisor of 501)