What are the divisors of 2045?

1, 5, 409, 2045

There is a total of 4 positive divisors.
The sum of these divisors is 2460.
The arithmetic mean is 615.

4 odd divisors

1, 5, 409, 2045

How to compute the divisors of 2045?

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 2045 by each of the numbers from 1 to 2045 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)

2045 / 1 = 2045 (the remainder is 0, so 1 is a divisor of 2045)
2045 / 2 = 1022.5 (the remainder is 1, so 2 is not a divisor of 2045)
2045 / 3 = 681.66666666667 (the remainder is 2, so 3 is not a divisor of 2045)
...
2045 / 2044 = 1.0004892367906 (the remainder is 1, so 2044 is not a divisor of 2045)
2045 / 2045 = 1 (the remainder is 0, so 2045 is a divisor of 2045)

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 2045 (i.e. 45.22167621838). 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 $\frac{N}{d} = D$ )

2045 / 1 = 2045 (the remainder is 0, so 1 and 2045 are divisors of 2045)
2045 / 2 = 1022.5 (the remainder is 1, so 2 is not a divisor of 2045)
2045 / 3 = 681.66666666667 (the remainder is 2, so 3 is not a divisor of 2045)
...
2045 / 44 = 46.477272727273 (the remainder is 21, so 44 is not a divisor of 2045)
2045 / 45 = 45.444444444444 (the remainder is 20, so 45 is not a divisor of 2045)