What are the divisors of 3599?

1, 59, 61, 3599

There is a total of 4 positive divisors.
The sum of these divisors is 3720.
The arithmetic mean is 930.

4 odd divisors

1, 59, 61, 3599

How to compute the divisors of 3599?

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 3599 by each of the numbers from 1 to 3599 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)

3599 / 1 = 3599 (the remainder is 0, so 1 is a divisor of 3599)
3599 / 2 = 1799.5 (the remainder is 1, so 2 is not a divisor of 3599)
3599 / 3 = 1199.6666666667 (the remainder is 2, so 3 is not a divisor of 3599)
...
3599 / 3598 = 1.0002779321845 (the remainder is 1, so 3598 is not a divisor of 3599)
3599 / 3599 = 1 (the remainder is 0, so 3599 is a divisor of 3599)

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 3599 (i.e. 59.991666087883). 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 $\frac{N}{d} = D$ )

3599 / 1 = 3599 (the remainder is 0, so 1 and 3599 are divisors of 3599)
3599 / 2 = 1799.5 (the remainder is 1, so 2 is not a divisor of 3599)
3599 / 3 = 1199.6666666667 (the remainder is 2, so 3 is not a divisor of 3599)
...
3599 / 58 = 62.051724137931 (the remainder is 3, so 58 is not a divisor of 3599)
3599 / 59 = 61 (the remainder is 0, so 59 and 61 are divisors of 3599)