What are the divisors of 4038?

1, 2, 3, 6, 673, 1346, 2019, 4038

There is a total of 8 positive divisors.
The sum of these divisors is 8088.
The arithmetic mean is 1011.

4 even divisors

2, 6, 1346, 4038

4 odd divisors

1, 3, 673, 2019

How to compute the divisors of 4038?

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 4038 by each of the numbers from 1 to 4038 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)

4038 / 1 = 4038 (the remainder is 0, so 1 is a divisor of 4038)
4038 / 2 = 2019 (the remainder is 0, so 2 is a divisor of 4038)
4038 / 3 = 1346 (the remainder is 0, so 3 is a divisor of 4038)
...
4038 / 4037 = 1.0002477086946 (the remainder is 1, so 4037 is not a divisor of 4038)
4038 / 4038 = 1 (the remainder is 0, so 4038 is a divisor of 4038)

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 4038 (i.e. 63.545259461269). 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 $\frac{N}{d} = D$ )

4038 / 1 = 4038 (the remainder is 0, so 1 and 4038 are divisors of 4038)
4038 / 2 = 2019 (the remainder is 0, so 2 and 2019 are divisors of 4038)
4038 / 3 = 1346 (the remainder is 0, so 3 and 1346 are divisors of 4038)
...
4038 / 62 = 65.129032258065 (the remainder is 8, so 62 is not a divisor of 4038)
4038 / 63 = 64.095238095238 (the remainder is 6, so 63 is not a divisor of 4038)