What are the divisors of 4517?

1, 4517

There is a total of 2 positive divisors.
The sum of these divisors is 4518.
The arithmetic mean is 2259.

2 odd divisors

1, 4517

How to compute the divisors of 4517?

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 4517 by each of the numbers from 1 to 4517 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)

4517 / 1 = 4517 (the remainder is 0, so 1 is a divisor of 4517)
4517 / 2 = 2258.5 (the remainder is 1, so 2 is not a divisor of 4517)
4517 / 3 = 1505.6666666667 (the remainder is 2, so 3 is not a divisor of 4517)
...
4517 / 4516 = 1.0002214348981 (the remainder is 1, so 4516 is not a divisor of 4517)
4517 / 4517 = 1 (the remainder is 0, so 4517 is a divisor of 4517)

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 4517 (i.e. 67.208630398186). 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 $\frac{N}{d} = D$ )

4517 / 1 = 4517 (the remainder is 0, so 1 and 4517 are divisors of 4517)
4517 / 2 = 2258.5 (the remainder is 1, so 2 is not a divisor of 4517)
4517 / 3 = 1505.6666666667 (the remainder is 2, so 3 is not a divisor of 4517)
...
4517 / 66 = 68.439393939394 (the remainder is 29, so 66 is not a divisor of 4517)
4517 / 67 = 67.417910447761 (the remainder is 28, so 67 is not a divisor of 4517)