What are the divisors of 513?

1, 3, 9, 19, 27, 57, 171, 513

There is a total of 8 positive divisors.
The sum of these divisors is 800.
The arithmetic mean is 100.

8 odd divisors

1, 3, 9, 19, 27, 57, 171, 513

How to compute the divisors of 513?

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 513 by each of the numbers from 1 to 513 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)

513 / 1 = 513 (the remainder is 0, so 1 is a divisor of 513)
513 / 2 = 256.5 (the remainder is 1, so 2 is not a divisor of 513)
513 / 3 = 171 (the remainder is 0, so 3 is a divisor of 513)
...
513 / 512 = 1.001953125 (the remainder is 1, so 512 is not a divisor of 513)
513 / 513 = 1 (the remainder is 0, so 513 is a divisor of 513)

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 513 (i.e. 22.649503305812). 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 $\frac{N}{d} = D$ )

513 / 1 = 513 (the remainder is 0, so 1 and 513 are divisors of 513)
513 / 2 = 256.5 (the remainder is 1, so 2 is not a divisor of 513)
513 / 3 = 171 (the remainder is 0, so 3 and 171 are divisors of 513)
...
513 / 21 = 24.428571428571 (the remainder is 9, so 21 is not a divisor of 513)
513 / 22 = 23.318181818182 (the remainder is 7, so 22 is not a divisor of 513)