What are the divisors of 8175?

1, 3, 5, 15, 25, 75, 109, 327, 545, 1635, 2725, 8175

There is a total of 12 positive divisors.
The sum of these divisors is 13640.
The arithmetic mean is 1136.6666666667.

12 odd divisors

1, 3, 5, 15, 25, 75, 109, 327, 545, 1635, 2725, 8175

How to compute the divisors of 8175?

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 8175 by each of the numbers from 1 to 8175 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)

8175 / 1 = 8175 (the remainder is 0, so 1 is a divisor of 8175)
8175 / 2 = 4087.5 (the remainder is 1, so 2 is not a divisor of 8175)
8175 / 3 = 2725 (the remainder is 0, so 3 is a divisor of 8175)
...
8175 / 8174 = 1.0001223391241 (the remainder is 1, so 8174 is not a divisor of 8175)
8175 / 8175 = 1 (the remainder is 0, so 8175 is a divisor of 8175)

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 8175 (i.e. 90.415706600126). 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 $\frac{N}{d} = D$ )

8175 / 1 = 8175 (the remainder is 0, so 1 and 8175 are divisors of 8175)
8175 / 2 = 4087.5 (the remainder is 1, so 2 is not a divisor of 8175)
8175 / 3 = 2725 (the remainder is 0, so 3 and 2725 are divisors of 8175)
...
8175 / 89 = 91.85393258427 (the remainder is 76, so 89 is not a divisor of 8175)
8175 / 90 = 90.833333333333 (the remainder is 75, so 90 is not a divisor of 8175)