What are the divisors of 524?

1, 2, 4, 131, 262, 524

There is a total of 6 positive divisors.
The sum of these divisors is 924.
The arithmetic mean is 154.

4 even divisors

2, 4, 262, 524

2 odd divisors

1, 131

How to compute the divisors of 524?

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 524 by each of the numbers from 1 to 524 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)

524 / 1 = 524 (the remainder is 0, so 1 is a divisor of 524)
524 / 2 = 262 (the remainder is 0, so 2 is a divisor of 524)
524 / 3 = 174.66666666667 (the remainder is 2, so 3 is not a divisor of 524)
...
524 / 523 = 1.0019120458891 (the remainder is 1, so 523 is not a divisor of 524)
524 / 524 = 1 (the remainder is 0, so 524 is a divisor of 524)

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 524 (i.e. 22.891046284519). 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 $\frac{N}{d} = D$ )

524 / 1 = 524 (the remainder is 0, so 1 and 524 are divisors of 524)
524 / 2 = 262 (the remainder is 0, so 2 and 262 are divisors of 524)
524 / 3 = 174.66666666667 (the remainder is 2, so 3 is not a divisor of 524)
...
524 / 21 = 24.952380952381 (the remainder is 20, so 21 is not a divisor of 524)
524 / 22 = 23.818181818182 (the remainder is 18, so 22 is not a divisor of 524)