What are the divisors of 3565?

1, 5, 23, 31, 115, 155, 713, 3565

There is a total of 8 positive divisors.
The sum of these divisors is 4608.
The arithmetic mean is 576.

8 odd divisors

1, 5, 23, 31, 115, 155, 713, 3565

How to compute the divisors of 3565?

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 3565 by each of the numbers from 1 to 3565 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)

3565 / 1 = 3565 (the remainder is 0, so 1 is a divisor of 3565)
3565 / 2 = 1782.5 (the remainder is 1, so 2 is not a divisor of 3565)
3565 / 3 = 1188.3333333333 (the remainder is 1, so 3 is not a divisor of 3565)
...
3565 / 3564 = 1.0002805836139 (the remainder is 1, so 3564 is not a divisor of 3565)
3565 / 3565 = 1 (the remainder is 0, so 3565 is a divisor of 3565)

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 3565 (i.e. 59.707620954113). 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 $\frac{N}{d} = D$ )

3565 / 1 = 3565 (the remainder is 0, so 1 and 3565 are divisors of 3565)
3565 / 2 = 1782.5 (the remainder is 1, so 2 is not a divisor of 3565)
3565 / 3 = 1188.3333333333 (the remainder is 1, so 3 is not a divisor of 3565)
...
3565 / 58 = 61.465517241379 (the remainder is 27, so 58 is not a divisor of 3565)
3565 / 59 = 60.423728813559 (the remainder is 25, so 59 is not a divisor of 3565)