What are the divisors of 4045?

1, 5, 809, 4045

There is a total of 4 positive divisors.
The sum of these divisors is 4860.
The arithmetic mean is 1215.

4 odd divisors

1, 5, 809, 4045

How to compute the divisors of 4045?

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 4045 by each of the numbers from 1 to 4045 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)

4045 / 1 = 4045 (the remainder is 0, so 1 is a divisor of 4045)
4045 / 2 = 2022.5 (the remainder is 1, so 2 is not a divisor of 4045)
4045 / 3 = 1348.3333333333 (the remainder is 1, so 3 is not a divisor of 4045)
...
4045 / 4044 = 1.0002472799209 (the remainder is 1, so 4044 is not a divisor of 4045)
4045 / 4045 = 1 (the remainder is 0, so 4045 is a divisor of 4045)

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 4045 (i.e. 63.600314464631). 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 $\frac{N}{d} = D$ )

4045 / 1 = 4045 (the remainder is 0, so 1 and 4045 are divisors of 4045)
4045 / 2 = 2022.5 (the remainder is 1, so 2 is not a divisor of 4045)
4045 / 3 = 1348.3333333333 (the remainder is 1, so 3 is not a divisor of 4045)
...
4045 / 62 = 65.241935483871 (the remainder is 15, so 62 is not a divisor of 4045)
4045 / 63 = 64.206349206349 (the remainder is 13, so 63 is not a divisor of 4045)