What are the divisors of 4195?

1, 5, 839, 4195

There is a total of 4 positive divisors.
The sum of these divisors is 5040.
The arithmetic mean is 1260.

4 odd divisors

1, 5, 839, 4195

How to compute the divisors of 4195?

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 4195 by each of the numbers from 1 to 4195 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)

4195 / 1 = 4195 (the remainder is 0, so 1 is a divisor of 4195)
4195 / 2 = 2097.5 (the remainder is 1, so 2 is not a divisor of 4195)
4195 / 3 = 1398.3333333333 (the remainder is 1, so 3 is not a divisor of 4195)
...
4195 / 4194 = 1.0002384358608 (the remainder is 1, so 4194 is not a divisor of 4195)
4195 / 4195 = 1 (the remainder is 0, so 4195 is a divisor of 4195)

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 4195 (i.e. 64.768819658845). 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 $\frac{N}{d} = D$ )

4195 / 1 = 4195 (the remainder is 0, so 1 and 4195 are divisors of 4195)
4195 / 2 = 2097.5 (the remainder is 1, so 2 is not a divisor of 4195)
4195 / 3 = 1398.3333333333 (the remainder is 1, so 3 is not a divisor of 4195)
...
4195 / 63 = 66.587301587302 (the remainder is 37, so 63 is not a divisor of 4195)
4195 / 64 = 65.546875 (the remainder is 35, so 64 is not a divisor of 4195)