What are the divisors of 3199?

1, 7, 457, 3199

There is a total of 4 positive divisors.
The sum of these divisors is 3664.
The arithmetic mean is 916.

4 odd divisors

1, 7, 457, 3199

How to compute the divisors of 3199?

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 3199 by each of the numbers from 1 to 3199 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)

3199 / 1 = 3199 (the remainder is 0, so 1 is a divisor of 3199)
3199 / 2 = 1599.5 (the remainder is 1, so 2 is not a divisor of 3199)
3199 / 3 = 1066.3333333333 (the remainder is 1, so 3 is not a divisor of 3199)
...
3199 / 3198 = 1.0003126954346 (the remainder is 1, so 3198 is not a divisor of 3199)
3199 / 3199 = 1 (the remainder is 0, so 3199 is a divisor of 3199)

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 3199 (i.e. 56.559702969517). 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 $\frac{N}{d} = D$ )

3199 / 1 = 3199 (the remainder is 0, so 1 and 3199 are divisors of 3199)
3199 / 2 = 1599.5 (the remainder is 1, so 2 is not a divisor of 3199)
3199 / 3 = 1066.3333333333 (the remainder is 1, so 3 is not a divisor of 3199)
...
3199 / 55 = 58.163636363636 (the remainder is 9, so 55 is not a divisor of 3199)
3199 / 56 = 57.125 (the remainder is 7, so 56 is not a divisor of 3199)