What are the divisors of 2041?

1, 13, 157, 2041

There is a total of 4 positive divisors.
The sum of these divisors is 2212.
The arithmetic mean is 553.

4 odd divisors

1, 13, 157, 2041

How to compute the divisors of 2041?

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 2041 by each of the numbers from 1 to 2041 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)

2041 / 1 = 2041 (the remainder is 0, so 1 is a divisor of 2041)
2041 / 2 = 1020.5 (the remainder is 1, so 2 is not a divisor of 2041)
2041 / 3 = 680.33333333333 (the remainder is 1, so 3 is not a divisor of 2041)
...
2041 / 2040 = 1.0004901960784 (the remainder is 1, so 2040 is not a divisor of 2041)
2041 / 2041 = 1 (the remainder is 0, so 2041 is a divisor of 2041)

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 2041 (i.e. 45.177427992306). 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 $\frac{N}{d} = D$ )

2041 / 1 = 2041 (the remainder is 0, so 1 and 2041 are divisors of 2041)
2041 / 2 = 1020.5 (the remainder is 1, so 2 is not a divisor of 2041)
2041 / 3 = 680.33333333333 (the remainder is 1, so 3 is not a divisor of 2041)
...
2041 / 44 = 46.386363636364 (the remainder is 17, so 44 is not a divisor of 2041)
2041 / 45 = 45.355555555556 (the remainder is 16, so 45 is not a divisor of 2041)