What are the divisors of 5214?

1, 2, 3, 6, 11, 22, 33, 66, 79, 158, 237, 474, 869, 1738, 2607, 5214

There is a total of 16 positive divisors.
The sum of these divisors is 11520.
The arithmetic mean is 720.

8 even divisors

2, 6, 22, 66, 158, 474, 1738, 5214

8 odd divisors

1, 3, 11, 33, 79, 237, 869, 2607

How to compute the divisors of 5214?

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 5214 by each of the numbers from 1 to 5214 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)

5214 / 1 = 5214 (the remainder is 0, so 1 is a divisor of 5214)
5214 / 2 = 2607 (the remainder is 0, so 2 is a divisor of 5214)
5214 / 3 = 1738 (the remainder is 0, so 3 is a divisor of 5214)
...
5214 / 5213 = 1.000191828122 (the remainder is 1, so 5213 is not a divisor of 5214)
5214 / 5214 = 1 (the remainder is 0, so 5214 is a divisor of 5214)

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 5214 (i.e. 72.208032794143). 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 $\frac{N}{d} = D$ )

5214 / 1 = 5214 (the remainder is 0, so 1 and 5214 are divisors of 5214)
5214 / 2 = 2607 (the remainder is 0, so 2 and 2607 are divisors of 5214)
5214 / 3 = 1738 (the remainder is 0, so 3 and 1738 are divisors of 5214)
...
5214 / 71 = 73.43661971831 (the remainder is 31, so 71 is not a divisor of 5214)
5214 / 72 = 72.416666666667 (the remainder is 30, so 72 is not a divisor of 5214)