What are the divisors of 5119?

1, 5119

There is a total of 2 positive divisors.
The sum of these divisors is 5120.
The arithmetic mean is 2560.

2 odd divisors

1, 5119

How to compute the divisors of 5119?

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 5119 by each of the numbers from 1 to 5119 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)

5119 / 1 = 5119 (the remainder is 0, so 1 is a divisor of 5119)
5119 / 2 = 2559.5 (the remainder is 1, so 2 is not a divisor of 5119)
5119 / 3 = 1706.3333333333 (the remainder is 1, so 3 is not a divisor of 5119)
...
5119 / 5118 = 1.0001953888238 (the remainder is 1, so 5118 is not a divisor of 5119)
5119 / 5119 = 1 (the remainder is 0, so 5119 is a divisor of 5119)

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 5119 (i.e. 71.547187226333). 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 $\frac{N}{d} = D$ )

5119 / 1 = 5119 (the remainder is 0, so 1 and 5119 are divisors of 5119)
5119 / 2 = 2559.5 (the remainder is 1, so 2 is not a divisor of 5119)
5119 / 3 = 1706.3333333333 (the remainder is 1, so 3 is not a divisor of 5119)
...
5119 / 70 = 73.128571428571 (the remainder is 9, so 70 is not a divisor of 5119)
5119 / 71 = 72.098591549296 (the remainder is 7, so 71 is not a divisor of 5119)