What are the divisors of 4500?

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 25, 30, 36, 45, 50, 60, 75, 90, 100, 125, 150, 180, 225, 250, 300, 375, 450, 500, 750, 900, 1125, 1500, 2250, 4500

There is a total of 36 positive divisors.
The sum of these divisors is 14196.
The arithmetic mean is 394.33333333333.

24 even divisors

2, 4, 6, 10, 12, 18, 20, 30, 36, 50, 60, 90, 100, 150, 180, 250, 300, 450, 500, 750, 900, 1500, 2250, 4500

12 odd divisors

1, 3, 5, 9, 15, 25, 45, 75, 125, 225, 375, 1125

How to compute the divisors of 4500?

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 4500 by each of the numbers from 1 to 4500 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)

4500 / 1 = 4500 (the remainder is 0, so 1 is a divisor of 4500)
4500 / 2 = 2250 (the remainder is 0, so 2 is a divisor of 4500)
4500 / 3 = 1500 (the remainder is 0, so 3 is a divisor of 4500)
...
4500 / 4499 = 1.0002222716159 (the remainder is 1, so 4499 is not a divisor of 4500)
4500 / 4500 = 1 (the remainder is 0, so 4500 is a divisor of 4500)

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 4500 (i.e. 67.082039324994). 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 $\frac{N}{d} = D$ )

4500 / 1 = 4500 (the remainder is 0, so 1 and 4500 are divisors of 4500)
4500 / 2 = 2250 (the remainder is 0, so 2 and 2250 are divisors of 4500)
4500 / 3 = 1500 (the remainder is 0, so 3 and 1500 are divisors of 4500)
...
4500 / 66 = 68.181818181818 (the remainder is 12, so 66 is not a divisor of 4500)
4500 / 67 = 67.164179104478 (the remainder is 11, so 67 is not a divisor of 4500)