What are the divisors of 4049?

1, 4049

There is a total of 2 positive divisors.
The sum of these divisors is 4050.
The arithmetic mean is 2025.

2 odd divisors

1, 4049

How to compute the divisors of 4049?

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 4049 by each of the numbers from 1 to 4049 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)

4049 / 1 = 4049 (the remainder is 0, so 1 is a divisor of 4049)
4049 / 2 = 2024.5 (the remainder is 1, so 2 is not a divisor of 4049)
4049 / 3 = 1349.6666666667 (the remainder is 2, so 3 is not a divisor of 4049)
...
4049 / 4048 = 1.0002470355731 (the remainder is 1, so 4048 is not a divisor of 4049)
4049 / 4049 = 1 (the remainder is 0, so 4049 is a divisor of 4049)

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 4049 (i.e. 63.631753079732). 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 $\frac{N}{d} = D$ )

4049 / 1 = 4049 (the remainder is 0, so 1 and 4049 are divisors of 4049)
4049 / 2 = 2024.5 (the remainder is 1, so 2 is not a divisor of 4049)
4049 / 3 = 1349.6666666667 (the remainder is 2, so 3 is not a divisor of 4049)
...
4049 / 62 = 65.306451612903 (the remainder is 19, so 62 is not a divisor of 4049)
4049 / 63 = 64.269841269841 (the remainder is 17, so 63 is not a divisor of 4049)