What are the divisors of 5213?

1, 13, 401, 5213

There is a total of 4 positive divisors.
The sum of these divisors is 5628.
The arithmetic mean is 1407.

4 odd divisors

1, 13, 401, 5213

How to compute the divisors of 5213?

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 5213 by each of the numbers from 1 to 5213 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)

5213 / 1 = 5213 (the remainder is 0, so 1 is a divisor of 5213)
5213 / 2 = 2606.5 (the remainder is 1, so 2 is not a divisor of 5213)
5213 / 3 = 1737.6666666667 (the remainder is 2, so 3 is not a divisor of 5213)
...
5213 / 5212 = 1.0001918649271 (the remainder is 1, so 5212 is not a divisor of 5213)
5213 / 5213 = 1 (the remainder is 0, so 5213 is a divisor of 5213)

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 5213 (i.e. 72.201108024739). 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 $\frac{N}{d} = D$ )

5213 / 1 = 5213 (the remainder is 0, so 1 and 5213 are divisors of 5213)
5213 / 2 = 2606.5 (the remainder is 1, so 2 is not a divisor of 5213)
5213 / 3 = 1737.6666666667 (the remainder is 2, so 3 is not a divisor of 5213)
...
5213 / 71 = 73.422535211268 (the remainder is 30, so 71 is not a divisor of 5213)
5213 / 72 = 72.402777777778 (the remainder is 29, so 72 is not a divisor of 5213)