What are the divisors of 5219?

1, 17, 307, 5219

There is a total of 4 positive divisors.
The sum of these divisors is 5544.
The arithmetic mean is 1386.

4 odd divisors

1, 17, 307, 5219

How to compute the divisors of 5219?

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 5219 by each of the numbers from 1 to 5219 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)

5219 / 1 = 5219 (the remainder is 0, so 1 is a divisor of 5219)
5219 / 2 = 2609.5 (the remainder is 1, so 2 is not a divisor of 5219)
5219 / 3 = 1739.6666666667 (the remainder is 2, so 3 is not a divisor of 5219)
...
5219 / 5218 = 1.0001916443082 (the remainder is 1, so 5218 is not a divisor of 5219)
5219 / 5219 = 1 (the remainder is 0, so 5219 is a divisor of 5219)

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 5219 (i.e. 72.242646684628). 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 $\frac{N}{d} = D$ )

5219 / 1 = 5219 (the remainder is 0, so 1 and 5219 are divisors of 5219)
5219 / 2 = 2609.5 (the remainder is 1, so 2 is not a divisor of 5219)
5219 / 3 = 1739.6666666667 (the remainder is 2, so 3 is not a divisor of 5219)
...
5219 / 71 = 73.507042253521 (the remainder is 36, so 71 is not a divisor of 5219)
5219 / 72 = 72.486111111111 (the remainder is 35, so 72 is not a divisor of 5219)