What are the divisors of 3209?

1, 3209

There is a total of 2 positive divisors.
The sum of these divisors is 3210.
The arithmetic mean is 1605.

2 odd divisors

1, 3209

How to compute the divisors of 3209?

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 3209 by each of the numbers from 1 to 3209 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)

3209 / 1 = 3209 (the remainder is 0, so 1 is a divisor of 3209)
3209 / 2 = 1604.5 (the remainder is 1, so 2 is not a divisor of 3209)
3209 / 3 = 1069.6666666667 (the remainder is 2, so 3 is not a divisor of 3209)
...
3209 / 3208 = 1.0003117206983 (the remainder is 1, so 3208 is not a divisor of 3209)
3209 / 3209 = 1 (the remainder is 0, so 3209 is a divisor of 3209)

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 3209 (i.e. 56.648036153074). 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 $\frac{N}{d} = D$ )

3209 / 1 = 3209 (the remainder is 0, so 1 and 3209 are divisors of 3209)
3209 / 2 = 1604.5 (the remainder is 1, so 2 is not a divisor of 3209)
3209 / 3 = 1069.6666666667 (the remainder is 2, so 3 is not a divisor of 3209)
...
3209 / 55 = 58.345454545455 (the remainder is 19, so 55 is not a divisor of 3209)
3209 / 56 = 57.303571428571 (the remainder is 17, so 56 is not a divisor of 3209)