What are the divisors of 2719?

1, 2719

There is a total of 2 positive divisors.
The sum of these divisors is 2720.
The arithmetic mean is 1360.

2 odd divisors

1, 2719

How to compute the divisors of 2719?

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 2719 by each of the numbers from 1 to 2719 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)

2719 / 1 = 2719 (the remainder is 0, so 1 is a divisor of 2719)
2719 / 2 = 1359.5 (the remainder is 1, so 2 is not a divisor of 2719)
2719 / 3 = 906.33333333333 (the remainder is 1, so 3 is not a divisor of 2719)
...
2719 / 2718 = 1.0003679175865 (the remainder is 1, so 2718 is not a divisor of 2719)
2719 / 2719 = 1 (the remainder is 0, so 2719 is a divisor of 2719)

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 2719 (i.e. 52.144031297935). 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 $\frac{N}{d} = D$ )

2719 / 1 = 2719 (the remainder is 0, so 1 and 2719 are divisors of 2719)
2719 / 2 = 1359.5 (the remainder is 1, so 2 is not a divisor of 2719)
2719 / 3 = 906.33333333333 (the remainder is 1, so 3 is not a divisor of 2719)
...
2719 / 51 = 53.313725490196 (the remainder is 16, so 51 is not a divisor of 2719)
2719 / 52 = 52.288461538462 (the remainder is 15, so 52 is not a divisor of 2719)