What are the divisors of 4075?

1, 5, 25, 163, 815, 4075

There is a total of 6 positive divisors.
The sum of these divisors is 5084.
The arithmetic mean is 847.33333333333.

6 odd divisors

1, 5, 25, 163, 815, 4075

How to compute the divisors of 4075?

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 4075 by each of the numbers from 1 to 4075 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)

4075 / 1 = 4075 (the remainder is 0, so 1 is a divisor of 4075)
4075 / 2 = 2037.5 (the remainder is 1, so 2 is not a divisor of 4075)
4075 / 3 = 1358.3333333333 (the remainder is 1, so 3 is not a divisor of 4075)
...
4075 / 4074 = 1.0002454590083 (the remainder is 1, so 4074 is not a divisor of 4075)
4075 / 4075 = 1 (the remainder is 0, so 4075 is a divisor of 4075)

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 4075 (i.e. 63.835726674019). 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 $\frac{N}{d} = D$ )

4075 / 1 = 4075 (the remainder is 0, so 1 and 4075 are divisors of 4075)
4075 / 2 = 2037.5 (the remainder is 1, so 2 is not a divisor of 4075)
4075 / 3 = 1358.3333333333 (the remainder is 1, so 3 is not a divisor of 4075)
...
4075 / 62 = 65.725806451613 (the remainder is 45, so 62 is not a divisor of 4075)
4075 / 63 = 64.68253968254 (the remainder is 43, so 63 is not a divisor of 4075)