What are the divisors of 3563?

1, 7, 509, 3563

There is a total of 4 positive divisors.
The sum of these divisors is 4080.
The arithmetic mean is 1020.

4 odd divisors

1, 7, 509, 3563

How to compute the divisors of 3563?

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 3563 by each of the numbers from 1 to 3563 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)

3563 / 1 = 3563 (the remainder is 0, so 1 is a divisor of 3563)
3563 / 2 = 1781.5 (the remainder is 1, so 2 is not a divisor of 3563)
3563 / 3 = 1187.6666666667 (the remainder is 2, so 3 is not a divisor of 3563)
...
3563 / 3562 = 1.0002807411567 (the remainder is 1, so 3562 is not a divisor of 3563)
3563 / 3563 = 1 (the remainder is 0, so 3563 is a divisor of 3563)

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 3563 (i.e. 59.690870323694). 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 $\frac{N}{d} = D$ )

3563 / 1 = 3563 (the remainder is 0, so 1 and 3563 are divisors of 3563)
3563 / 2 = 1781.5 (the remainder is 1, so 2 is not a divisor of 3563)
3563 / 3 = 1187.6666666667 (the remainder is 2, so 3 is not a divisor of 3563)
...
3563 / 58 = 61.431034482759 (the remainder is 25, so 58 is not a divisor of 3563)
3563 / 59 = 60.389830508475 (the remainder is 23, so 59 is not a divisor of 3563)