What are the divisors of 1609?

1, 1609

There is a total of 2 positive divisors.
The sum of these divisors is 1610.
The arithmetic mean is 805.

2 odd divisors

1, 1609

How to compute the divisors of 1609?

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 1609 by each of the numbers from 1 to 1609 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)

1609 / 1 = 1609 (the remainder is 0, so 1 is a divisor of 1609)
1609 / 2 = 804.5 (the remainder is 1, so 2 is not a divisor of 1609)
1609 / 3 = 536.33333333333 (the remainder is 1, so 3 is not a divisor of 1609)
...
1609 / 1608 = 1.0006218905473 (the remainder is 1, so 1608 is not a divisor of 1609)
1609 / 1609 = 1 (the remainder is 0, so 1609 is a divisor of 1609)

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 1609 (i.e. 40.112342240263). 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 $\frac{N}{d} = D$ )

1609 / 1 = 1609 (the remainder is 0, so 1 and 1609 are divisors of 1609)
1609 / 2 = 804.5 (the remainder is 1, so 2 is not a divisor of 1609)
1609 / 3 = 536.33333333333 (the remainder is 1, so 3 is not a divisor of 1609)
...
1609 / 39 = 41.25641025641 (the remainder is 10, so 39 is not a divisor of 1609)
1609 / 40 = 40.225 (the remainder is 9, so 40 is not a divisor of 1609)