What are the divisors of 9075?

1, 3, 5, 11, 15, 25, 33, 55, 75, 121, 165, 275, 363, 605, 825, 1815, 3025, 9075

There is a total of 18 positive divisors.
The sum of these divisors is 16492.
The arithmetic mean is 916.22222222222.

18 odd divisors

1, 3, 5, 11, 15, 25, 33, 55, 75, 121, 165, 275, 363, 605, 825, 1815, 3025, 9075

How to compute the divisors of 9075?

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 9075 by each of the numbers from 1 to 9075 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)

9075 / 1 = 9075 (the remainder is 0, so 1 is a divisor of 9075)
9075 / 2 = 4537.5 (the remainder is 1, so 2 is not a divisor of 9075)
9075 / 3 = 3025 (the remainder is 0, so 3 is a divisor of 9075)
...
9075 / 9074 = 1.0001102049813 (the remainder is 1, so 9074 is not a divisor of 9075)
9075 / 9075 = 1 (the remainder is 0, so 9075 is a divisor of 9075)

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 9075 (i.e. 95.262794416288). 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 $\frac{N}{d} = D$ )

9075 / 1 = 9075 (the remainder is 0, so 1 and 9075 are divisors of 9075)
9075 / 2 = 4537.5 (the remainder is 1, so 2 is not a divisor of 9075)
9075 / 3 = 3025 (the remainder is 0, so 3 and 3025 are divisors of 9075)
...
9075 / 94 = 96.542553191489 (the remainder is 51, so 94 is not a divisor of 9075)
9075 / 95 = 95.526315789474 (the remainder is 50, so 95 is not a divisor of 9075)