What are the divisors of 9225?

1, 3, 5, 9, 15, 25, 41, 45, 75, 123, 205, 225, 369, 615, 1025, 1845, 3075, 9225

There is a total of 18 positive divisors.
The sum of these divisors is 16926.
The arithmetic mean is 940.33333333333.

18 odd divisors

1, 3, 5, 9, 15, 25, 41, 45, 75, 123, 205, 225, 369, 615, 1025, 1845, 3075, 9225

How to compute the divisors of 9225?

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 9225 by each of the numbers from 1 to 9225 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)

9225 / 1 = 9225 (the remainder is 0, so 1 is a divisor of 9225)
9225 / 2 = 4612.5 (the remainder is 1, so 2 is not a divisor of 9225)
9225 / 3 = 3075 (the remainder is 0, so 3 is a divisor of 9225)
...
9225 / 9224 = 1.0001084128361 (the remainder is 1, so 9224 is not a divisor of 9225)
9225 / 9225 = 1 (the remainder is 0, so 9225 is a divisor of 9225)

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 9225 (i.e. 96.046863561493). 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 $\frac{N}{d} = D$ )

9225 / 1 = 9225 (the remainder is 0, so 1 and 9225 are divisors of 9225)
9225 / 2 = 4612.5 (the remainder is 1, so 2 is not a divisor of 9225)
9225 / 3 = 3075 (the remainder is 0, so 3 and 3075 are divisors of 9225)
...
9225 / 95 = 97.105263157895 (the remainder is 10, so 95 is not a divisor of 9225)
9225 / 96 = 96.09375 (the remainder is 9, so 96 is not a divisor of 9225)