What are the divisors of 9000?

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 25, 30, 36, 40, 45, 50, 60, 72, 75, 90, 100, 120, 125, 150, 180, 200, 225, 250, 300, 360, 375, 450, 500, 600, 750, 900, 1000, 1125, 1500, 1800, 2250, 3000, 4500, 9000

There is a total of 48 positive divisors.
The sum of these divisors is 30420.
The arithmetic mean is 633.75.

36 even divisors

2, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 40, 50, 60, 72, 90, 100, 120, 150, 180, 200, 250, 300, 360, 450, 500, 600, 750, 900, 1000, 1500, 1800, 2250, 3000, 4500, 9000

12 odd divisors

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

How to compute the divisors of 9000?

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 9000 by each of the numbers from 1 to 9000 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)

9000 / 1 = 9000 (the remainder is 0, so 1 is a divisor of 9000)
9000 / 2 = 4500 (the remainder is 0, so 2 is a divisor of 9000)
9000 / 3 = 3000 (the remainder is 0, so 3 is a divisor of 9000)
...
9000 / 8999 = 1.0001111234582 (the remainder is 1, so 8999 is not a divisor of 9000)
9000 / 9000 = 1 (the remainder is 0, so 9000 is a divisor of 9000)

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 9000 (i.e. 94.868329805051). 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 $\frac{N}{d} = D$ )

9000 / 1 = 9000 (the remainder is 0, so 1 and 9000 are divisors of 9000)
9000 / 2 = 4500 (the remainder is 0, so 2 and 4500 are divisors of 9000)
9000 / 3 = 3000 (the remainder is 0, so 3 and 3000 are divisors of 9000)
...
9000 / 93 = 96.774193548387 (the remainder is 72, so 93 is not a divisor of 9000)
9000 / 94 = 95.744680851064 (the remainder is 70, so 94 is not a divisor of 9000)