What are the divisors of 8235?

1, 3, 5, 9, 15, 27, 45, 61, 135, 183, 305, 549, 915, 1647, 2745, 8235

There is a total of 16 positive divisors.
The sum of these divisors is 14880.
The arithmetic mean is 930.

16 odd divisors

1, 3, 5, 9, 15, 27, 45, 61, 135, 183, 305, 549, 915, 1647, 2745, 8235

How to compute the divisors of 8235?

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 8235 by each of the numbers from 1 to 8235 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)

8235 / 1 = 8235 (the remainder is 0, so 1 is a divisor of 8235)
8235 / 2 = 4117.5 (the remainder is 1, so 2 is not a divisor of 8235)
8235 / 3 = 2745 (the remainder is 0, so 3 is a divisor of 8235)
...
8235 / 8234 = 1.0001214476561 (the remainder is 1, so 8234 is not a divisor of 8235)
8235 / 8235 = 1 (the remainder is 0, so 8235 is a divisor of 8235)

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 8235 (i.e. 90.746900773525). 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 $\frac{N}{d} = D$ )

8235 / 1 = 8235 (the remainder is 0, so 1 and 8235 are divisors of 8235)
8235 / 2 = 4117.5 (the remainder is 1, so 2 is not a divisor of 8235)
8235 / 3 = 2745 (the remainder is 0, so 3 and 2745 are divisors of 8235)
...
8235 / 89 = 92.52808988764 (the remainder is 47, so 89 is not a divisor of 8235)
8235 / 90 = 91.5 (the remainder is 45, so 90 is not a divisor of 8235)