What are the divisors of 4523?

1, 4523

There is a total of 2 positive divisors.
The sum of these divisors is 4524.
The arithmetic mean is 2262.

2 odd divisors

1, 4523

How to compute the divisors of 4523?

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 4523 by each of the numbers from 1 to 4523 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)

4523 / 1 = 4523 (the remainder is 0, so 1 is a divisor of 4523)
4523 / 2 = 2261.5 (the remainder is 1, so 2 is not a divisor of 4523)
4523 / 3 = 1507.6666666667 (the remainder is 2, so 3 is not a divisor of 4523)
...
4523 / 4522 = 1.000221141088 (the remainder is 1, so 4522 is not a divisor of 4523)
4523 / 4523 = 1 (the remainder is 0, so 4523 is a divisor of 4523)

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 4523 (i.e. 67.253252709441). 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 $\frac{N}{d} = D$ )

4523 / 1 = 4523 (the remainder is 0, so 1 and 4523 are divisors of 4523)
4523 / 2 = 2261.5 (the remainder is 1, so 2 is not a divisor of 4523)
4523 / 3 = 1507.6666666667 (the remainder is 2, so 3 is not a divisor of 4523)
...
4523 / 66 = 68.530303030303 (the remainder is 35, so 66 is not a divisor of 4523)
4523 / 67 = 67.507462686567 (the remainder is 34, so 67 is not a divisor of 4523)