What are the divisors of 675?

1, 3, 5, 9, 15, 25, 27, 45, 75, 135, 225, 675

There is a total of 12 positive divisors.
The sum of these divisors is 1240.
The arithmetic mean is 103.33333333333.

12 odd divisors

1, 3, 5, 9, 15, 25, 27, 45, 75, 135, 225, 675

How to compute the divisors of 675?

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 675 by each of the numbers from 1 to 675 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)

675 / 1 = 675 (the remainder is 0, so 1 is a divisor of 675)
675 / 2 = 337.5 (the remainder is 1, so 2 is not a divisor of 675)
675 / 3 = 225 (the remainder is 0, so 3 is a divisor of 675)
...
675 / 674 = 1.0014836795252 (the remainder is 1, so 674 is not a divisor of 675)
675 / 675 = 1 (the remainder is 0, so 675 is a divisor of 675)

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 675 (i.e. 25.980762113533). 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 $\frac{N}{d} = D$ )

675 / 1 = 675 (the remainder is 0, so 1 and 675 are divisors of 675)
675 / 2 = 337.5 (the remainder is 1, so 2 is not a divisor of 675)
675 / 3 = 225 (the remainder is 0, so 3 and 225 are divisors of 675)
...
675 / 24 = 28.125 (the remainder is 3, so 24 is not a divisor of 675)
675 / 25 = 27 (the remainder is 0, so 25 and 27 are divisors of 675)