What are the divisors of 1214?

1, 2, 607, 1214

There is a total of 4 positive divisors.
The sum of these divisors is 1824.
The arithmetic mean is 456.

2 even divisors

2, 1214

2 odd divisors

1, 607

How to compute the divisors of 1214?

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 1214 by each of the numbers from 1 to 1214 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)

1214 / 1 = 1214 (the remainder is 0, so 1 is a divisor of 1214)
1214 / 2 = 607 (the remainder is 0, so 2 is a divisor of 1214)
1214 / 3 = 404.66666666667 (the remainder is 2, so 3 is not a divisor of 1214)
...
1214 / 1213 = 1.0008244023083 (the remainder is 1, so 1213 is not a divisor of 1214)
1214 / 1214 = 1 (the remainder is 0, so 1214 is a divisor of 1214)

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 1214 (i.e. 34.842502780369). 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 $\frac{N}{d} = D$ )

1214 / 1 = 1214 (the remainder is 0, so 1 and 1214 are divisors of 1214)
1214 / 2 = 607 (the remainder is 0, so 2 and 607 are divisors of 1214)
1214 / 3 = 404.66666666667 (the remainder is 2, so 3 is not a divisor of 1214)
...
1214 / 33 = 36.787878787879 (the remainder is 26, so 33 is not a divisor of 1214)
1214 / 34 = 35.705882352941 (the remainder is 24, so 34 is not a divisor of 1214)