What are the divisors of 521?

1, 521

There is a total of 2 positive divisors.
The sum of these divisors is 522.
The arithmetic mean is 261.

2 odd divisors

1, 521

How to compute the divisors of 521?

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 521 by each of the numbers from 1 to 521 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)

521 / 1 = 521 (the remainder is 0, so 1 is a divisor of 521)
521 / 2 = 260.5 (the remainder is 1, so 2 is not a divisor of 521)
521 / 3 = 173.66666666667 (the remainder is 2, so 3 is not a divisor of 521)
...
521 / 520 = 1.0019230769231 (the remainder is 1, so 520 is not a divisor of 521)
521 / 521 = 1 (the remainder is 0, so 521 is a divisor of 521)

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 521 (i.e. 22.825424421027). 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 $\frac{N}{d} = D$ )

521 / 1 = 521 (the remainder is 0, so 1 and 521 are divisors of 521)
521 / 2 = 260.5 (the remainder is 1, so 2 is not a divisor of 521)
521 / 3 = 173.66666666667 (the remainder is 2, so 3 is not a divisor of 521)
...
521 / 21 = 24.809523809524 (the remainder is 17, so 21 is not a divisor of 521)
521 / 22 = 23.681818181818 (the remainder is 15, so 22 is not a divisor of 521)