What are the divisors of 2131?

1, 2131

There is a total of 2 positive divisors.
The sum of these divisors is 2132.
The arithmetic mean is 1066.

2 odd divisors

1, 2131

How to compute the divisors of 2131?

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 2131 by each of the numbers from 1 to 2131 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)

2131 / 1 = 2131 (the remainder is 0, so 1 is a divisor of 2131)
2131 / 2 = 1065.5 (the remainder is 1, so 2 is not a divisor of 2131)
2131 / 3 = 710.33333333333 (the remainder is 1, so 3 is not a divisor of 2131)
...
2131 / 2130 = 1.0004694835681 (the remainder is 1, so 2130 is not a divisor of 2131)
2131 / 2131 = 1 (the remainder is 0, so 2131 is a divisor of 2131)

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 2131 (i.e. 46.162755550335). 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 $\frac{N}{d} = D$ )

2131 / 1 = 2131 (the remainder is 0, so 1 and 2131 are divisors of 2131)
2131 / 2 = 1065.5 (the remainder is 1, so 2 is not a divisor of 2131)
2131 / 3 = 710.33333333333 (the remainder is 1, so 3 is not a divisor of 2131)
...
2131 / 45 = 47.355555555556 (the remainder is 16, so 45 is not a divisor of 2131)
2131 / 46 = 46.326086956522 (the remainder is 15, so 46 is not a divisor of 2131)