What are the divisors of 4563?

1, 3, 9, 13, 27, 39, 117, 169, 351, 507, 1521, 4563

There is a total of 12 positive divisors.
The sum of these divisors is 7320.
The arithmetic mean is 610.

12 odd divisors

1, 3, 9, 13, 27, 39, 117, 169, 351, 507, 1521, 4563

How to compute the divisors of 4563?

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 4563 by each of the numbers from 1 to 4563 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)

4563 / 1 = 4563 (the remainder is 0, so 1 is a divisor of 4563)
4563 / 2 = 2281.5 (the remainder is 1, so 2 is not a divisor of 4563)
4563 / 3 = 1521 (the remainder is 0, so 3 is a divisor of 4563)
...
4563 / 4562 = 1.0002192021043 (the remainder is 1, so 4562 is not a divisor of 4563)
4563 / 4563 = 1 (the remainder is 0, so 4563 is a divisor of 4563)

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 4563 (i.e. 67.549981495186). 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 $\frac{N}{d} = D$ )

4563 / 1 = 4563 (the remainder is 0, so 1 and 4563 are divisors of 4563)
4563 / 2 = 2281.5 (the remainder is 1, so 2 is not a divisor of 4563)
4563 / 3 = 1521 (the remainder is 0, so 3 and 1521 are divisors of 4563)
...
4563 / 66 = 69.136363636364 (the remainder is 9, so 66 is not a divisor of 4563)
4563 / 67 = 68.10447761194 (the remainder is 7, so 67 is not a divisor of 4563)