What are the divisors of 3191?

1, 3191

There is a total of 2 positive divisors.
The sum of these divisors is 3192.
The arithmetic mean is 1596.

2 odd divisors

1, 3191

How to compute the divisors of 3191?

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 3191 by each of the numbers from 1 to 3191 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)

3191 / 1 = 3191 (the remainder is 0, so 1 is a divisor of 3191)
3191 / 2 = 1595.5 (the remainder is 1, so 2 is not a divisor of 3191)
3191 / 3 = 1063.6666666667 (the remainder is 2, so 3 is not a divisor of 3191)
...
3191 / 3190 = 1.0003134796238 (the remainder is 1, so 3190 is not a divisor of 3191)
3191 / 3191 = 1 (the remainder is 0, so 3191 is a divisor of 3191)

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 3191 (i.e. 56.488936969994). 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 $\frac{N}{d} = D$ )

3191 / 1 = 3191 (the remainder is 0, so 1 and 3191 are divisors of 3191)
3191 / 2 = 1595.5 (the remainder is 1, so 2 is not a divisor of 3191)
3191 / 3 = 1063.6666666667 (the remainder is 2, so 3 is not a divisor of 3191)
...
3191 / 55 = 58.018181818182 (the remainder is 1, so 55 is not a divisor of 3191)
3191 / 56 = 56.982142857143 (the remainder is 55, so 56 is not a divisor of 3191)