What are the divisors of 3000?

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 25, 30, 40, 50, 60, 75, 100, 120, 125, 150, 200, 250, 300, 375, 500, 600, 750, 1000, 1500, 3000

There is a total of 32 positive divisors.
The sum of these divisors is 9360.
The arithmetic mean is 292.5.

24 even divisors

2, 4, 6, 8, 10, 12, 20, 24, 30, 40, 50, 60, 100, 120, 150, 200, 250, 300, 500, 600, 750, 1000, 1500, 3000

8 odd divisors

1, 3, 5, 15, 25, 75, 125, 375

How to compute the divisors of 3000?

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 3000 by each of the numbers from 1 to 3000 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)

3000 / 1 = 3000 (the remainder is 0, so 1 is a divisor of 3000)
3000 / 2 = 1500 (the remainder is 0, so 2 is a divisor of 3000)
3000 / 3 = 1000 (the remainder is 0, so 3 is a divisor of 3000)
...
3000 / 2999 = 1.0003334444815 (the remainder is 1, so 2999 is not a divisor of 3000)
3000 / 3000 = 1 (the remainder is 0, so 3000 is a divisor of 3000)

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 3000 (i.e. 54.772255750517). 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 $\frac{N}{d} = D$ )

3000 / 1 = 3000 (the remainder is 0, so 1 and 3000 are divisors of 3000)
3000 / 2 = 1500 (the remainder is 0, so 2 and 1500 are divisors of 3000)
3000 / 3 = 1000 (the remainder is 0, so 3 and 1000 are divisors of 3000)
...
3000 / 53 = 56.603773584906 (the remainder is 32, so 53 is not a divisor of 3000)
3000 / 54 = 55.555555555556 (the remainder is 30, so 54 is not a divisor of 3000)