What are the divisors of 1709?

1, 1709

There is a total of 2 positive divisors.
The sum of these divisors is 1710.
The arithmetic mean is 855.

2 odd divisors

1, 1709

How to compute the divisors of 1709?

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 1709 by each of the numbers from 1 to 1709 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)

1709 / 1 = 1709 (the remainder is 0, so 1 is a divisor of 1709)
1709 / 2 = 854.5 (the remainder is 1, so 2 is not a divisor of 1709)
1709 / 3 = 569.66666666667 (the remainder is 2, so 3 is not a divisor of 1709)
...
1709 / 1708 = 1.0005854800937 (the remainder is 1, so 1708 is not a divisor of 1709)
1709 / 1709 = 1 (the remainder is 0, so 1709 is a divisor of 1709)

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 1709 (i.e. 41.340053217189). 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 $\frac{N}{d} = D$ )

1709 / 1 = 1709 (the remainder is 0, so 1 and 1709 are divisors of 1709)
1709 / 2 = 854.5 (the remainder is 1, so 2 is not a divisor of 1709)
1709 / 3 = 569.66666666667 (the remainder is 2, so 3 is not a divisor of 1709)
...
1709 / 40 = 42.725 (the remainder is 29, so 40 is not a divisor of 1709)
1709 / 41 = 41.682926829268 (the remainder is 28, so 41 is not a divisor of 1709)