What are the divisors of 4021?

1, 4021

There is a total of 2 positive divisors.
The sum of these divisors is 4022.
The arithmetic mean is 2011.

2 odd divisors

1, 4021

How to compute the divisors of 4021?

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 4021 by each of the numbers from 1 to 4021 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)

4021 / 1 = 4021 (the remainder is 0, so 1 is a divisor of 4021)
4021 / 2 = 2010.5 (the remainder is 1, so 2 is not a divisor of 4021)
4021 / 3 = 1340.3333333333 (the remainder is 1, so 3 is not a divisor of 4021)
...
4021 / 4020 = 1.0002487562189 (the remainder is 1, so 4020 is not a divisor of 4021)
4021 / 4021 = 1 (the remainder is 0, so 4021 is a divisor of 4021)

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 4021 (i.e. 63.411355449951). 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 $\frac{N}{d} = D$ )

4021 / 1 = 4021 (the remainder is 0, so 1 and 4021 are divisors of 4021)
4021 / 2 = 2010.5 (the remainder is 1, so 2 is not a divisor of 4021)
4021 / 3 = 1340.3333333333 (the remainder is 1, so 3 is not a divisor of 4021)
...
4021 / 62 = 64.854838709677 (the remainder is 53, so 62 is not a divisor of 4021)
4021 / 63 = 63.825396825397 (the remainder is 52, so 63 is not a divisor of 4021)