What are the divisors of 2014?

1, 2, 19, 38, 53, 106, 1007, 2014

There is a total of 8 positive divisors.
The sum of these divisors is 3240.
The arithmetic mean is 405.

4 even divisors

2, 38, 106, 2014

4 odd divisors

1, 19, 53, 1007

How to compute the divisors of 2014?

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 2014 by each of the numbers from 1 to 2014 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)

2014 / 1 = 2014 (the remainder is 0, so 1 is a divisor of 2014)
2014 / 2 = 1007 (the remainder is 0, so 2 is a divisor of 2014)
2014 / 3 = 671.33333333333 (the remainder is 1, so 3 is not a divisor of 2014)
...
2014 / 2013 = 1.0004967709886 (the remainder is 1, so 2013 is not a divisor of 2014)
2014 / 2014 = 1 (the remainder is 0, so 2014 is a divisor of 2014)

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 2014 (i.e. 44.877611344634). 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 $\frac{N}{d} = D$ )

2014 / 1 = 2014 (the remainder is 0, so 1 and 2014 are divisors of 2014)
2014 / 2 = 1007 (the remainder is 0, so 2 and 1007 are divisors of 2014)
2014 / 3 = 671.33333333333 (the remainder is 1, so 3 is not a divisor of 2014)
...
2014 / 43 = 46.837209302326 (the remainder is 36, so 43 is not a divisor of 2014)
2014 / 44 = 45.772727272727 (the remainder is 34, so 44 is not a divisor of 2014)