What are the divisors of 214?

1, 2, 107, 214

There is a total of 4 positive divisors.
The sum of these divisors is 324.
The arithmetic mean is 81.

2 even divisors

2, 214

2 odd divisors

1, 107

How to compute the divisors of 214?

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 214 by each of the numbers from 1 to 214 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)

214 / 1 = 214 (the remainder is 0, so 1 is a divisor of 214)
214 / 2 = 107 (the remainder is 0, so 2 is a divisor of 214)
214 / 3 = 71.333333333333 (the remainder is 1, so 3 is not a divisor of 214)
...
214 / 213 = 1.0046948356808 (the remainder is 1, so 213 is not a divisor of 214)
214 / 214 = 1 (the remainder is 0, so 214 is a divisor of 214)

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 214 (i.e. 14.628738838328). 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 $\frac{N}{d} = D$ )

214 / 1 = 214 (the remainder is 0, so 1 and 214 are divisors of 214)
214 / 2 = 107 (the remainder is 0, so 2 and 107 are divisors of 214)
214 / 3 = 71.333333333333 (the remainder is 1, so 3 is not a divisor of 214)
...
214 / 13 = 16.461538461538 (the remainder is 6, so 13 is not a divisor of 214)
214 / 14 = 15.285714285714 (the remainder is 4, so 14 is not a divisor of 214)