What are the divisors of 4035?

1, 3, 5, 15, 269, 807, 1345, 4035

There is a total of 8 positive divisors.
The sum of these divisors is 6480.
The arithmetic mean is 810.

8 odd divisors

1, 3, 5, 15, 269, 807, 1345, 4035

How to compute the divisors of 4035?

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 4035 by each of the numbers from 1 to 4035 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)

4035 / 1 = 4035 (the remainder is 0, so 1 is a divisor of 4035)
4035 / 2 = 2017.5 (the remainder is 1, so 2 is not a divisor of 4035)
4035 / 3 = 1345 (the remainder is 0, so 3 is a divisor of 4035)
...
4035 / 4034 = 1.0002478929103 (the remainder is 1, so 4034 is not a divisor of 4035)
4035 / 4035 = 1 (the remainder is 0, so 4035 is a divisor of 4035)

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 4035 (i.e. 63.521649852629). 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 $\frac{N}{d} = D$ )

4035 / 1 = 4035 (the remainder is 0, so 1 and 4035 are divisors of 4035)
4035 / 2 = 2017.5 (the remainder is 1, so 2 is not a divisor of 4035)
4035 / 3 = 1345 (the remainder is 0, so 3 and 1345 are divisors of 4035)
...
4035 / 62 = 65.08064516129 (the remainder is 5, so 62 is not a divisor of 4035)
4035 / 63 = 64.047619047619 (the remainder is 3, so 63 is not a divisor of 4035)