What are the divisors of 4069?

1, 13, 313, 4069

There is a total of 4 positive divisors.
The sum of these divisors is 4396.
The arithmetic mean is 1099.

4 odd divisors

1, 13, 313, 4069

How to compute the divisors of 4069?

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 4069 by each of the numbers from 1 to 4069 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)

4069 / 1 = 4069 (the remainder is 0, so 1 is a divisor of 4069)
4069 / 2 = 2034.5 (the remainder is 1, so 2 is not a divisor of 4069)
4069 / 3 = 1356.3333333333 (the remainder is 1, so 3 is not a divisor of 4069)
...
4069 / 4068 = 1.0002458210423 (the remainder is 1, so 4068 is not a divisor of 4069)
4069 / 4069 = 1 (the remainder is 0, so 4069 is a divisor of 4069)

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 4069 (i.e. 63.788713735268). 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 $\frac{N}{d} = D$ )

4069 / 1 = 4069 (the remainder is 0, so 1 and 4069 are divisors of 4069)
4069 / 2 = 2034.5 (the remainder is 1, so 2 is not a divisor of 4069)
4069 / 3 = 1356.3333333333 (the remainder is 1, so 3 is not a divisor of 4069)
...
4069 / 62 = 65.629032258065 (the remainder is 39, so 62 is not a divisor of 4069)
4069 / 63 = 64.587301587302 (the remainder is 37, so 63 is not a divisor of 4069)