What are the divisors of 538?

1, 2, 269, 538

There is a total of 4 positive divisors.
The sum of these divisors is 810.
The arithmetic mean is 202.5.

2 even divisors

2, 538

2 odd divisors

1, 269

How to compute the divisors of 538?

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 538 by each of the numbers from 1 to 538 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)

538 / 1 = 538 (the remainder is 0, so 1 is a divisor of 538)
538 / 2 = 269 (the remainder is 0, so 2 is a divisor of 538)
538 / 3 = 179.33333333333 (the remainder is 1, so 3 is not a divisor of 538)
...
538 / 537 = 1.0018621973929 (the remainder is 1, so 537 is not a divisor of 538)
538 / 538 = 1 (the remainder is 0, so 538 is a divisor of 538)

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 538 (i.e. 23.194827009486). 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 $\frac{N}{d} = D$ )

538 / 1 = 538 (the remainder is 0, so 1 and 538 are divisors of 538)
538 / 2 = 269 (the remainder is 0, so 2 and 269 are divisors of 538)
538 / 3 = 179.33333333333 (the remainder is 1, so 3 is not a divisor of 538)
...
538 / 22 = 24.454545454545 (the remainder is 10, so 22 is not a divisor of 538)
538 / 23 = 23.391304347826 (the remainder is 9, so 23 is not a divisor of 538)