What are the divisors of 2731?

1, 2731

There is a total of 2 positive divisors.
The sum of these divisors is 2732.
The arithmetic mean is 1366.

2 odd divisors

1, 2731

How to compute the divisors of 2731?

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 2731 by each of the numbers from 1 to 2731 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)

2731 / 1 = 2731 (the remainder is 0, so 1 is a divisor of 2731)
2731 / 2 = 1365.5 (the remainder is 1, so 2 is not a divisor of 2731)
2731 / 3 = 910.33333333333 (the remainder is 1, so 3 is not a divisor of 2731)
...
2731 / 2730 = 1.0003663003663 (the remainder is 1, so 2730 is not a divisor of 2731)
2731 / 2731 = 1 (the remainder is 0, so 2731 is a divisor of 2731)

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 2731 (i.e. 52.258970521816). 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 $\frac{N}{d} = D$ )

2731 / 1 = 2731 (the remainder is 0, so 1 and 2731 are divisors of 2731)
2731 / 2 = 1365.5 (the remainder is 1, so 2 is not a divisor of 2731)
2731 / 3 = 910.33333333333 (the remainder is 1, so 3 is not a divisor of 2731)
...
2731 / 51 = 53.549019607843 (the remainder is 28, so 51 is not a divisor of 2731)
2731 / 52 = 52.519230769231 (the remainder is 27, so 52 is not a divisor of 2731)