What are the divisors of 225?

1, 3, 5, 9, 15, 25, 45, 75, 225

There is a total of 9 positive divisors.
The sum of these divisors is 403.
The arithmetic mean is 44.777777777778.

9 odd divisors

1, 3, 5, 9, 15, 25, 45, 75, 225

How to compute the divisors of 225?

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 225 by each of the numbers from 1 to 225 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)

225 / 1 = 225 (the remainder is 0, so 1 is a divisor of 225)
225 / 2 = 112.5 (the remainder is 1, so 2 is not a divisor of 225)
225 / 3 = 75 (the remainder is 0, so 3 is a divisor of 225)
...
225 / 224 = 1.0044642857143 (the remainder is 1, so 224 is not a divisor of 225)
225 / 225 = 1 (the remainder is 0, so 225 is a divisor of 225)

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 225 (i.e. 15). 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 $\frac{N}{d} = D$ )

225 / 1 = 225 (the remainder is 0, so 1 and 225 are divisors of 225)
225 / 2 = 112.5 (the remainder is 1, so 2 is not a divisor of 225)
225 / 3 = 75 (the remainder is 0, so 3 and 75 are divisors of 225)
...
225 / 14 = 16.071428571429 (the remainder is 1, so 14 is not a divisor of 225)
225 / 15 = 15 (the remainder is 0, so 15 and 15 are divisors of 225)