What are the divisors of 527?

1, 17, 31, 527

There is a total of 4 positive divisors.
The sum of these divisors is 576.
The arithmetic mean is 144.

4 odd divisors

1, 17, 31, 527

How to compute the divisors of 527?

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 527 by each of the numbers from 1 to 527 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)

527 / 1 = 527 (the remainder is 0, so 1 is a divisor of 527)
527 / 2 = 263.5 (the remainder is 1, so 2 is not a divisor of 527)
527 / 3 = 175.66666666667 (the remainder is 2, so 3 is not a divisor of 527)
...
527 / 526 = 1.0019011406844 (the remainder is 1, so 526 is not a divisor of 527)
527 / 527 = 1 (the remainder is 0, so 527 is a divisor of 527)

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 527 (i.e. 22.956480566498). 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 $\frac{N}{d} = D$ )

527 / 1 = 527 (the remainder is 0, so 1 and 527 are divisors of 527)
527 / 2 = 263.5 (the remainder is 1, so 2 is not a divisor of 527)
527 / 3 = 175.66666666667 (the remainder is 2, so 3 is not a divisor of 527)
...
527 / 21 = 25.095238095238 (the remainder is 2, so 21 is not a divisor of 527)
527 / 22 = 23.954545454545 (the remainder is 21, so 22 is not a divisor of 527)