What are the divisors of 2214?

1, 2, 3, 6, 9, 18, 27, 41, 54, 82, 123, 246, 369, 738, 1107, 2214

There is a total of 16 positive divisors.
The sum of these divisors is 5040.
The arithmetic mean is 315.

8 even divisors

2, 6, 18, 54, 82, 246, 738, 2214

8 odd divisors

1, 3, 9, 27, 41, 123, 369, 1107

How to compute the divisors of 2214?

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 2214 by each of the numbers from 1 to 2214 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)

2214 / 1 = 2214 (the remainder is 0, so 1 is a divisor of 2214)
2214 / 2 = 1107 (the remainder is 0, so 2 is a divisor of 2214)
2214 / 3 = 738 (the remainder is 0, so 3 is a divisor of 2214)
...
2214 / 2213 = 1.0004518752824 (the remainder is 1, so 2213 is not a divisor of 2214)
2214 / 2214 = 1 (the remainder is 0, so 2214 is a divisor of 2214)

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 2214 (i.e. 47.053161424074). 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 $\frac{N}{d} = D$ )

2214 / 1 = 2214 (the remainder is 0, so 1 and 2214 are divisors of 2214)
2214 / 2 = 1107 (the remainder is 0, so 2 and 1107 are divisors of 2214)
2214 / 3 = 738 (the remainder is 0, so 3 and 738 are divisors of 2214)
...
2214 / 46 = 48.130434782609 (the remainder is 6, so 46 is not a divisor of 2214)
2214 / 47 = 47.106382978723 (the remainder is 5, so 47 is not a divisor of 2214)