What are the divisors of 2071?

1, 19, 109, 2071

There is a total of 4 positive divisors.
The sum of these divisors is 2200.
The arithmetic mean is 550.

4 odd divisors

1, 19, 109, 2071

How to compute the divisors of 2071?

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 2071 by each of the numbers from 1 to 2071 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)

2071 / 1 = 2071 (the remainder is 0, so 1 is a divisor of 2071)
2071 / 2 = 1035.5 (the remainder is 1, so 2 is not a divisor of 2071)
2071 / 3 = 690.33333333333 (the remainder is 1, so 3 is not a divisor of 2071)
...
2071 / 2070 = 1.0004830917874 (the remainder is 1, so 2070 is not a divisor of 2071)
2071 / 2071 = 1 (the remainder is 0, so 2071 is a divisor of 2071)

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 2071 (i.e. 45.508241011931). 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 $\frac{N}{d} = D$ )

2071 / 1 = 2071 (the remainder is 0, so 1 and 2071 are divisors of 2071)
2071 / 2 = 1035.5 (the remainder is 1, so 2 is not a divisor of 2071)
2071 / 3 = 690.33333333333 (the remainder is 1, so 3 is not a divisor of 2071)
...
2071 / 44 = 47.068181818182 (the remainder is 3, so 44 is not a divisor of 2071)
2071 / 45 = 46.022222222222 (the remainder is 1, so 45 is not a divisor of 2071)