What are the divisors of 3409?

1, 7, 487, 3409

There is a total of 4 positive divisors.
The sum of these divisors is 3904.
The arithmetic mean is 976.

4 odd divisors

1, 7, 487, 3409

How to compute the divisors of 3409?

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 3409 by each of the numbers from 1 to 3409 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)

3409 / 1 = 3409 (the remainder is 0, so 1 is a divisor of 3409)
3409 / 2 = 1704.5 (the remainder is 1, so 2 is not a divisor of 3409)
3409 / 3 = 1136.3333333333 (the remainder is 1, so 3 is not a divisor of 3409)
...
3409 / 3408 = 1.00029342723 (the remainder is 1, so 3408 is not a divisor of 3409)
3409 / 3409 = 1 (the remainder is 0, so 3409 is a divisor of 3409)

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 3409 (i.e. 58.38664230798). 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 $\frac{N}{d} = D$ )

3409 / 1 = 3409 (the remainder is 0, so 1 and 3409 are divisors of 3409)
3409 / 2 = 1704.5 (the remainder is 1, so 2 is not a divisor of 3409)
3409 / 3 = 1136.3333333333 (the remainder is 1, so 3 is not a divisor of 3409)
...
3409 / 57 = 59.80701754386 (the remainder is 46, so 57 is not a divisor of 3409)
3409 / 58 = 58.775862068966 (the remainder is 45, so 58 is not a divisor of 3409)