What are the divisors of 5191?

1, 29, 179, 5191

There is a total of 4 positive divisors.
The sum of these divisors is 5400.
The arithmetic mean is 1350.

4 odd divisors

1, 29, 179, 5191

How to compute the divisors of 5191?

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 5191 by each of the numbers from 1 to 5191 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)

5191 / 1 = 5191 (the remainder is 0, so 1 is a divisor of 5191)
5191 / 2 = 2595.5 (the remainder is 1, so 2 is not a divisor of 5191)
5191 / 3 = 1730.3333333333 (the remainder is 1, so 3 is not a divisor of 5191)
...
5191 / 5190 = 1.0001926782274 (the remainder is 1, so 5190 is not a divisor of 5191)
5191 / 5191 = 1 (the remainder is 0, so 5191 is a divisor of 5191)

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 5191 (i.e. 72.04859471218). 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 $\frac{N}{d} = D$ )

5191 / 1 = 5191 (the remainder is 0, so 1 and 5191 are divisors of 5191)
5191 / 2 = 2595.5 (the remainder is 1, so 2 is not a divisor of 5191)
5191 / 3 = 1730.3333333333 (the remainder is 1, so 3 is not a divisor of 5191)
...
5191 / 71 = 73.112676056338 (the remainder is 8, so 71 is not a divisor of 5191)
5191 / 72 = 72.097222222222 (the remainder is 7, so 72 is not a divisor of 5191)