What are the divisors of 8125?

1, 5, 13, 25, 65, 125, 325, 625, 1625, 8125

There is a total of 10 positive divisors.
The sum of these divisors is 10934.
The arithmetic mean is 1093.4.

10 odd divisors

1, 5, 13, 25, 65, 125, 325, 625, 1625, 8125

How to compute the divisors of 8125?

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 8125 by each of the numbers from 1 to 8125 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)

8125 / 1 = 8125 (the remainder is 0, so 1 is a divisor of 8125)
8125 / 2 = 4062.5 (the remainder is 1, so 2 is not a divisor of 8125)
8125 / 3 = 2708.3333333333 (the remainder is 1, so 3 is not a divisor of 8125)
...
8125 / 8124 = 1.0001230920729 (the remainder is 1, so 8124 is not a divisor of 8125)
8125 / 8125 = 1 (the remainder is 0, so 8125 is a divisor of 8125)

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 8125 (i.e. 90.1387818866). 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 $\frac{N}{d} = D$ )

8125 / 1 = 8125 (the remainder is 0, so 1 and 8125 are divisors of 8125)
8125 / 2 = 4062.5 (the remainder is 1, so 2 is not a divisor of 8125)
8125 / 3 = 2708.3333333333 (the remainder is 1, so 3 is not a divisor of 8125)
...
8125 / 89 = 91.292134831461 (the remainder is 26, so 89 is not a divisor of 8125)
8125 / 90 = 90.277777777778 (the remainder is 25, so 90 is not a divisor of 8125)