What are the divisors of 2231?

1, 23, 97, 2231

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

4 odd divisors

1, 23, 97, 2231

How to compute the divisors of 2231?

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 2231 by each of the numbers from 1 to 2231 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)

2231 / 1 = 2231 (the remainder is 0, so 1 is a divisor of 2231)
2231 / 2 = 1115.5 (the remainder is 1, so 2 is not a divisor of 2231)
2231 / 3 = 743.66666666667 (the remainder is 2, so 3 is not a divisor of 2231)
...
2231 / 2230 = 1.0004484304933 (the remainder is 1, so 2230 is not a divisor of 2231)
2231 / 2231 = 1 (the remainder is 0, so 2231 is a divisor of 2231)

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 2231 (i.e. 47.233462714478). 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 $\frac{N}{d} = D$ )

2231 / 1 = 2231 (the remainder is 0, so 1 and 2231 are divisors of 2231)
2231 / 2 = 1115.5 (the remainder is 1, so 2 is not a divisor of 2231)
2231 / 3 = 743.66666666667 (the remainder is 2, so 3 is not a divisor of 2231)
...
2231 / 46 = 48.5 (the remainder is 23, so 46 is not a divisor of 2231)
2231 / 47 = 47.468085106383 (the remainder is 22, so 47 is not a divisor of 2231)