What are the divisors of 4027?

1, 4027

There is a total of 2 positive divisors.
The sum of these divisors is 4028.
The arithmetic mean is 2014.

2 odd divisors

1, 4027

How to compute the divisors of 4027?

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 4027 by each of the numbers from 1 to 4027 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)

4027 / 1 = 4027 (the remainder is 0, so 1 is a divisor of 4027)
4027 / 2 = 2013.5 (the remainder is 1, so 2 is not a divisor of 4027)
4027 / 3 = 1342.3333333333 (the remainder is 1, so 3 is not a divisor of 4027)
...
4027 / 4026 = 1.0002483854943 (the remainder is 1, so 4026 is not a divisor of 4027)
4027 / 4027 = 1 (the remainder is 0, so 4027 is a divisor of 4027)

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 4027 (i.e. 63.45864795282). 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 $\frac{N}{d} = D$ )

4027 / 1 = 4027 (the remainder is 0, so 1 and 4027 are divisors of 4027)
4027 / 2 = 2013.5 (the remainder is 1, so 2 is not a divisor of 4027)
4027 / 3 = 1342.3333333333 (the remainder is 1, so 3 is not a divisor of 4027)
...
4027 / 62 = 64.951612903226 (the remainder is 59, so 62 is not a divisor of 4027)
4027 / 63 = 63.920634920635 (the remainder is 58, so 63 is not a divisor of 4027)