What are the divisors of 1593?

1, 3, 9, 27, 59, 177, 531, 1593

There is a total of 8 positive divisors.
The sum of these divisors is 2400.
The arithmetic mean is 300.

8 odd divisors

1, 3, 9, 27, 59, 177, 531, 1593

How to compute the divisors of 1593?

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 1593 by each of the numbers from 1 to 1593 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)

1593 / 1 = 1593 (the remainder is 0, so 1 is a divisor of 1593)
1593 / 2 = 796.5 (the remainder is 1, so 2 is not a divisor of 1593)
1593 / 3 = 531 (the remainder is 0, so 3 is a divisor of 1593)
...
1593 / 1592 = 1.0006281407035 (the remainder is 1, so 1592 is not a divisor of 1593)
1593 / 1593 = 1 (the remainder is 0, so 1593 is a divisor of 1593)

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 1593 (i.e. 39.91240408695). 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 $\frac{N}{d} = D$ )

1593 / 1 = 1593 (the remainder is 0, so 1 and 1593 are divisors of 1593)
1593 / 2 = 796.5 (the remainder is 1, so 2 is not a divisor of 1593)
1593 / 3 = 531 (the remainder is 0, so 3 and 531 are divisors of 1593)
...
1593 / 38 = 41.921052631579 (the remainder is 35, so 38 is not a divisor of 1593)
1593 / 39 = 40.846153846154 (the remainder is 33, so 39 is not a divisor of 1593)