What are the divisors of 4135?

1, 5, 827, 4135

There is a total of 4 positive divisors.
The sum of these divisors is 4968.
The arithmetic mean is 1242.

4 odd divisors

1, 5, 827, 4135

How to compute the divisors of 4135?

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 4135 by each of the numbers from 1 to 4135 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)

4135 / 1 = 4135 (the remainder is 0, so 1 is a divisor of 4135)
4135 / 2 = 2067.5 (the remainder is 1, so 2 is not a divisor of 4135)
4135 / 3 = 1378.3333333333 (the remainder is 1, so 3 is not a divisor of 4135)
...
4135 / 4134 = 1.0002418964683 (the remainder is 1, so 4134 is not a divisor of 4135)
4135 / 4135 = 1 (the remainder is 0, so 4135 is a divisor of 4135)

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 4135 (i.e. 64.303965663091). 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 $\frac{N}{d} = D$ )

4135 / 1 = 4135 (the remainder is 0, so 1 and 4135 are divisors of 4135)
4135 / 2 = 2067.5 (the remainder is 1, so 2 is not a divisor of 4135)
4135 / 3 = 1378.3333333333 (the remainder is 1, so 3 is not a divisor of 4135)
...
4135 / 63 = 65.634920634921 (the remainder is 40, so 63 is not a divisor of 4135)
4135 / 64 = 64.609375 (the remainder is 39, so 64 is not a divisor of 4135)