What are the divisors of 515?

1, 5, 103, 515

There is a total of 4 positive divisors.
The sum of these divisors is 624.
The arithmetic mean is 156.

4 odd divisors

1, 5, 103, 515

How to compute the divisors of 515?

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 515 by each of the numbers from 1 to 515 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)

515 / 1 = 515 (the remainder is 0, so 1 is a divisor of 515)
515 / 2 = 257.5 (the remainder is 1, so 2 is not a divisor of 515)
515 / 3 = 171.66666666667 (the remainder is 2, so 3 is not a divisor of 515)
...
515 / 514 = 1.0019455252918 (the remainder is 1, so 514 is not a divisor of 515)
515 / 515 = 1 (the remainder is 0, so 515 is a divisor of 515)

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 515 (i.e. 22.69361143582). 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 $\frac{N}{d} = D$ )

515 / 1 = 515 (the remainder is 0, so 1 and 515 are divisors of 515)
515 / 2 = 257.5 (the remainder is 1, so 2 is not a divisor of 515)
515 / 3 = 171.66666666667 (the remainder is 2, so 3 is not a divisor of 515)
...
515 / 21 = 24.52380952381 (the remainder is 11, so 21 is not a divisor of 515)
515 / 22 = 23.409090909091 (the remainder is 9, so 22 is not a divisor of 515)