What are the divisors of 4013?

1, 4013

There is a total of 2 positive divisors.
The sum of these divisors is 4014.
The arithmetic mean is 2007.

2 odd divisors

1, 4013

How to compute the divisors of 4013?

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 4013 by each of the numbers from 1 to 4013 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)

4013 / 1 = 4013 (the remainder is 0, so 1 is a divisor of 4013)
4013 / 2 = 2006.5 (the remainder is 1, so 2 is not a divisor of 4013)
4013 / 3 = 1337.6666666667 (the remainder is 2, so 3 is not a divisor of 4013)
...
4013 / 4012 = 1.0002492522433 (the remainder is 1, so 4012 is not a divisor of 4013)
4013 / 4013 = 1 (the remainder is 0, so 4013 is a divisor of 4013)

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 4013 (i.e. 63.348243858847). 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 $\frac{N}{d} = D$ )

4013 / 1 = 4013 (the remainder is 0, so 1 and 4013 are divisors of 4013)
4013 / 2 = 2006.5 (the remainder is 1, so 2 is not a divisor of 4013)
4013 / 3 = 1337.6666666667 (the remainder is 2, so 3 is not a divisor of 4013)
...
4013 / 62 = 64.725806451613 (the remainder is 45, so 62 is not a divisor of 4013)
4013 / 63 = 63.698412698413 (the remainder is 44, so 63 is not a divisor of 4013)