What are the divisors of 8225?

1, 5, 7, 25, 35, 47, 175, 235, 329, 1175, 1645, 8225

There is a total of 12 positive divisors.
The sum of these divisors is 11904.
The arithmetic mean is 992.

12 odd divisors

1, 5, 7, 25, 35, 47, 175, 235, 329, 1175, 1645, 8225

How to compute the divisors of 8225?

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 8225 by each of the numbers from 1 to 8225 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)

8225 / 1 = 8225 (the remainder is 0, so 1 is a divisor of 8225)
8225 / 2 = 4112.5 (the remainder is 1, so 2 is not a divisor of 8225)
8225 / 3 = 2741.6666666667 (the remainder is 2, so 3 is not a divisor of 8225)
...
8225 / 8224 = 1.0001215953307 (the remainder is 1, so 8224 is not a divisor of 8225)
8225 / 8225 = 1 (the remainder is 0, so 8225 is a divisor of 8225)

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 8225 (i.e. 90.691785736085). 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 $\frac{N}{d} = D$ )

8225 / 1 = 8225 (the remainder is 0, so 1 and 8225 are divisors of 8225)
8225 / 2 = 4112.5 (the remainder is 1, so 2 is not a divisor of 8225)
8225 / 3 = 2741.6666666667 (the remainder is 2, so 3 is not a divisor of 8225)
...
8225 / 89 = 92.415730337079 (the remainder is 37, so 89 is not a divisor of 8225)
8225 / 90 = 91.388888888889 (the remainder is 35, so 90 is not a divisor of 8225)