What are the divisors of 4305?

1, 3, 5, 7, 15, 21, 35, 41, 105, 123, 205, 287, 615, 861, 1435, 4305

There is a total of 16 positive divisors.
The sum of these divisors is 8064.
The arithmetic mean is 504.

16 odd divisors

1, 3, 5, 7, 15, 21, 35, 41, 105, 123, 205, 287, 615, 861, 1435, 4305

How to compute the divisors of 4305?

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 4305 by each of the numbers from 1 to 4305 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)

4305 / 1 = 4305 (the remainder is 0, so 1 is a divisor of 4305)
4305 / 2 = 2152.5 (the remainder is 1, so 2 is not a divisor of 4305)
4305 / 3 = 1435 (the remainder is 0, so 3 is a divisor of 4305)
...
4305 / 4304 = 1.0002323420074 (the remainder is 1, so 4304 is not a divisor of 4305)
4305 / 4305 = 1 (the remainder is 0, so 4305 is a divisor of 4305)

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 4305 (i.e. 65.612498809297). 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 $\frac{N}{d} = D$ )

4305 / 1 = 4305 (the remainder is 0, so 1 and 4305 are divisors of 4305)
4305 / 2 = 2152.5 (the remainder is 1, so 2 is not a divisor of 4305)
4305 / 3 = 1435 (the remainder is 0, so 3 and 1435 are divisors of 4305)
...
4305 / 64 = 67.265625 (the remainder is 17, so 64 is not a divisor of 4305)
4305 / 65 = 66.230769230769 (the remainder is 15, so 65 is not a divisor of 4305)