What are the divisors of 2699?

1, 2699

There is a total of 2 positive divisors.
The sum of these divisors is 2700.
The arithmetic mean is 1350.

2 odd divisors

1, 2699

How to compute the divisors of 2699?

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 2699 by each of the numbers from 1 to 2699 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)

2699 / 1 = 2699 (the remainder is 0, so 1 is a divisor of 2699)
2699 / 2 = 1349.5 (the remainder is 1, so 2 is not a divisor of 2699)
2699 / 3 = 899.66666666667 (the remainder is 2, so 3 is not a divisor of 2699)
...
2699 / 2698 = 1.0003706449222 (the remainder is 1, so 2698 is not a divisor of 2699)
2699 / 2699 = 1 (the remainder is 0, so 2699 is a divisor of 2699)

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 2699 (i.e. 51.951900831442). 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 $\frac{N}{d} = D$ )

2699 / 1 = 2699 (the remainder is 0, so 1 and 2699 are divisors of 2699)
2699 / 2 = 1349.5 (the remainder is 1, so 2 is not a divisor of 2699)
2699 / 3 = 899.66666666667 (the remainder is 2, so 3 is not a divisor of 2699)
...
2699 / 50 = 53.98 (the remainder is 49, so 50 is not a divisor of 2699)
2699 / 51 = 52.921568627451 (the remainder is 47, so 51 is not a divisor of 2699)