What are the divisors of 4365?

1, 3, 5, 9, 15, 45, 97, 291, 485, 873, 1455, 4365

There is a total of 12 positive divisors.
The sum of these divisors is 7644.
The arithmetic mean is 637.

12 odd divisors

1, 3, 5, 9, 15, 45, 97, 291, 485, 873, 1455, 4365

How to compute the divisors of 4365?

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 4365 by each of the numbers from 1 to 4365 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)

4365 / 1 = 4365 (the remainder is 0, so 1 is a divisor of 4365)
4365 / 2 = 2182.5 (the remainder is 1, so 2 is not a divisor of 4365)
4365 / 3 = 1455 (the remainder is 0, so 3 is a divisor of 4365)
...
4365 / 4364 = 1.000229147571 (the remainder is 1, so 4364 is not a divisor of 4365)
4365 / 4365 = 1 (the remainder is 0, so 4365 is a divisor of 4365)

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 4365 (i.e. 66.068146636636). 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 $\frac{N}{d} = D$ )

4365 / 1 = 4365 (the remainder is 0, so 1 and 4365 are divisors of 4365)
4365 / 2 = 2182.5 (the remainder is 1, so 2 is not a divisor of 4365)
4365 / 3 = 1455 (the remainder is 0, so 3 and 1455 are divisors of 4365)
...
4365 / 65 = 67.153846153846 (the remainder is 10, so 65 is not a divisor of 4365)
4365 / 66 = 66.136363636364 (the remainder is 9, so 66 is not a divisor of 4365)