What are the divisors of 4515?

1, 3, 5, 7, 15, 21, 35, 43, 105, 129, 215, 301, 645, 903, 1505, 4515

There is a total of 16 positive divisors.
The sum of these divisors is 8448.
The arithmetic mean is 528.

16 odd divisors

1, 3, 5, 7, 15, 21, 35, 43, 105, 129, 215, 301, 645, 903, 1505, 4515

How to compute the divisors of 4515?

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 4515 by each of the numbers from 1 to 4515 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)

4515 / 1 = 4515 (the remainder is 0, so 1 is a divisor of 4515)
4515 / 2 = 2257.5 (the remainder is 1, so 2 is not a divisor of 4515)
4515 / 3 = 1505 (the remainder is 0, so 3 is a divisor of 4515)
...
4515 / 4514 = 1.0002215330084 (the remainder is 1, so 4514 is not a divisor of 4515)
4515 / 4515 = 1 (the remainder is 0, so 4515 is a divisor of 4515)

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 4515 (i.e. 67.193749709329). 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 $\frac{N}{d} = D$ )

4515 / 1 = 4515 (the remainder is 0, so 1 and 4515 are divisors of 4515)
4515 / 2 = 2257.5 (the remainder is 1, so 2 is not a divisor of 4515)
4515 / 3 = 1505 (the remainder is 0, so 3 and 1505 are divisors of 4515)
...
4515 / 66 = 68.409090909091 (the remainder is 27, so 66 is not a divisor of 4515)
4515 / 67 = 67.388059701493 (the remainder is 26, so 67 is not a divisor of 4515)