What are the divisors of 2051?

1, 7, 293, 2051

There is a total of 4 positive divisors.
The sum of these divisors is 2352.
The arithmetic mean is 588.

4 odd divisors

1, 7, 293, 2051

How to compute the divisors of 2051?

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 2051 by each of the numbers from 1 to 2051 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)

2051 / 1 = 2051 (the remainder is 0, so 1 is a divisor of 2051)
2051 / 2 = 1025.5 (the remainder is 1, so 2 is not a divisor of 2051)
2051 / 3 = 683.66666666667 (the remainder is 2, so 3 is not a divisor of 2051)
...
2051 / 2050 = 1.000487804878 (the remainder is 1, so 2050 is not a divisor of 2051)
2051 / 2051 = 1 (the remainder is 0, so 2051 is a divisor of 2051)

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 2051 (i.e. 45.287967496897). 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 $\frac{N}{d} = D$ )

2051 / 1 = 2051 (the remainder is 0, so 1 and 2051 are divisors of 2051)
2051 / 2 = 1025.5 (the remainder is 1, so 2 is not a divisor of 2051)
2051 / 3 = 683.66666666667 (the remainder is 2, so 3 is not a divisor of 2051)
...
2051 / 44 = 46.613636363636 (the remainder is 27, so 44 is not a divisor of 2051)
2051 / 45 = 45.577777777778 (the remainder is 26, so 45 is not a divisor of 2051)