What are the divisors of 2690?

1, 2, 5, 10, 269, 538, 1345, 2690

There is a total of 8 positive divisors.
The sum of these divisors is 4860.
The arithmetic mean is 607.5.

4 even divisors

2, 10, 538, 2690

4 odd divisors

1, 5, 269, 1345

How to compute the divisors of 2690?

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 2690 by each of the numbers from 1 to 2690 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)

2690 / 1 = 2690 (the remainder is 0, so 1 is a divisor of 2690)
2690 / 2 = 1345 (the remainder is 0, so 2 is a divisor of 2690)
2690 / 3 = 896.66666666667 (the remainder is 2, so 3 is not a divisor of 2690)
...
2690 / 2689 = 1.0003718854593 (the remainder is 1, so 2689 is not a divisor of 2690)
2690 / 2690 = 1 (the remainder is 0, so 2690 is a divisor of 2690)

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 2690 (i.e. 51.86520991956). 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 $\frac{N}{d} = D$ )

2690 / 1 = 2690 (the remainder is 0, so 1 and 2690 are divisors of 2690)
2690 / 2 = 1345 (the remainder is 0, so 2 and 1345 are divisors of 2690)
2690 / 3 = 896.66666666667 (the remainder is 2, so 3 is not a divisor of 2690)
...
2690 / 50 = 53.8 (the remainder is 40, so 50 is not a divisor of 2690)
2690 / 51 = 52.745098039216 (the remainder is 38, so 51 is not a divisor of 2690)