What are the divisors of 4214?

1, 2, 7, 14, 43, 49, 86, 98, 301, 602, 2107, 4214

There is a total of 12 positive divisors.
The sum of these divisors is 7524.
The arithmetic mean is 627.

6 even divisors

2, 14, 86, 98, 602, 4214

6 odd divisors

1, 7, 43, 49, 301, 2107

How to compute the divisors of 4214?

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 4214 by each of the numbers from 1 to 4214 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)

4214 / 1 = 4214 (the remainder is 0, so 1 is a divisor of 4214)
4214 / 2 = 2107 (the remainder is 0, so 2 is a divisor of 4214)
4214 / 3 = 1404.6666666667 (the remainder is 2, so 3 is not a divisor of 4214)
...
4214 / 4213 = 1.0002373605507 (the remainder is 1, so 4213 is not a divisor of 4214)
4214 / 4214 = 1 (the remainder is 0, so 4214 is a divisor of 4214)

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 4214 (i.e. 64.91532946847). 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 $\frac{N}{d} = D$ )

4214 / 1 = 4214 (the remainder is 0, so 1 and 4214 are divisors of 4214)
4214 / 2 = 2107 (the remainder is 0, so 2 and 2107 are divisors of 4214)
4214 / 3 = 1404.6666666667 (the remainder is 2, so 3 is not a divisor of 4214)
...
4214 / 63 = 66.888888888889 (the remainder is 56, so 63 is not a divisor of 4214)
4214 / 64 = 65.84375 (the remainder is 54, so 64 is not a divisor of 4214)