What are the divisors of 7035?

1, 3, 5, 7, 15, 21, 35, 67, 105, 201, 335, 469, 1005, 1407, 2345, 7035

There is a total of 16 positive divisors.
The sum of these divisors is 13056.
The arithmetic mean is 816.

16 odd divisors

1, 3, 5, 7, 15, 21, 35, 67, 105, 201, 335, 469, 1005, 1407, 2345, 7035

How to compute the divisors of 7035?

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 7035 by each of the numbers from 1 to 7035 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)

7035 / 1 = 7035 (the remainder is 0, so 1 is a divisor of 7035)
7035 / 2 = 3517.5 (the remainder is 1, so 2 is not a divisor of 7035)
7035 / 3 = 2345 (the remainder is 0, so 3 is a divisor of 7035)
...
7035 / 7034 = 1.0001421666193 (the remainder is 1, so 7034 is not a divisor of 7035)
7035 / 7035 = 1 (the remainder is 0, so 7035 is a divisor of 7035)

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 7035 (i.e. 83.874906855388). 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 $\frac{N}{d} = D$ )

7035 / 1 = 7035 (the remainder is 0, so 1 and 7035 are divisors of 7035)
7035 / 2 = 3517.5 (the remainder is 1, so 2 is not a divisor of 7035)
7035 / 3 = 2345 (the remainder is 0, so 3 and 2345 are divisors of 7035)
...
7035 / 82 = 85.792682926829 (the remainder is 65, so 82 is not a divisor of 7035)
7035 / 83 = 84.759036144578 (the remainder is 63, so 83 is not a divisor of 7035)