What are the divisors of 8025?

1, 3, 5, 15, 25, 75, 107, 321, 535, 1605, 2675, 8025

There is a total of 12 positive divisors.
The sum of these divisors is 13392.
The arithmetic mean is 1116.

12 odd divisors

1, 3, 5, 15, 25, 75, 107, 321, 535, 1605, 2675, 8025

How to compute the divisors of 8025?

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 8025 by each of the numbers from 1 to 8025 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)

8025 / 1 = 8025 (the remainder is 0, so 1 is a divisor of 8025)
8025 / 2 = 4012.5 (the remainder is 1, so 2 is not a divisor of 8025)
8025 / 3 = 2675 (the remainder is 0, so 3 is a divisor of 8025)
...
8025 / 8024 = 1.0001246261216 (the remainder is 1, so 8024 is not a divisor of 8025)
8025 / 8025 = 1 (the remainder is 0, so 8025 is a divisor of 8025)

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 8025 (i.e. 89.582364335845). 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 $\frac{N}{d} = D$ )

8025 / 1 = 8025 (the remainder is 0, so 1 and 8025 are divisors of 8025)
8025 / 2 = 4012.5 (the remainder is 1, so 2 is not a divisor of 8025)
8025 / 3 = 2675 (the remainder is 0, so 3 and 2675 are divisors of 8025)
...
8025 / 88 = 91.193181818182 (the remainder is 17, so 88 is not a divisor of 8025)
8025 / 89 = 90.168539325843 (the remainder is 15, so 89 is not a divisor of 8025)