What are the divisors of 5209?

1, 5209

There is a total of 2 positive divisors.
The sum of these divisors is 5210.
The arithmetic mean is 2605.

2 odd divisors

1, 5209

How to compute the divisors of 5209?

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 5209 by each of the numbers from 1 to 5209 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)

5209 / 1 = 5209 (the remainder is 0, so 1 is a divisor of 5209)
5209 / 2 = 2604.5 (the remainder is 1, so 2 is not a divisor of 5209)
5209 / 3 = 1736.3333333333 (the remainder is 1, so 3 is not a divisor of 5209)
...
5209 / 5208 = 1.0001920122888 (the remainder is 1, so 5208 is not a divisor of 5209)
5209 / 5209 = 1 (the remainder is 0, so 5209 is a divisor of 5209)

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 5209 (i.e. 72.173402303065). 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 $\frac{N}{d} = D$ )

5209 / 1 = 5209 (the remainder is 0, so 1 and 5209 are divisors of 5209)
5209 / 2 = 2604.5 (the remainder is 1, so 2 is not a divisor of 5209)
5209 / 3 = 1736.3333333333 (the remainder is 1, so 3 is not a divisor of 5209)
...
5209 / 71 = 73.366197183099 (the remainder is 26, so 71 is not a divisor of 5209)
5209 / 72 = 72.347222222222 (the remainder is 25, so 72 is not a divisor of 5209)