What are the divisors of 2698?

1, 2, 19, 38, 71, 142, 1349, 2698

There is a total of 8 positive divisors.
The sum of these divisors is 4320.
The arithmetic mean is 540.

4 even divisors

2, 38, 142, 2698

4 odd divisors

1, 19, 71, 1349

How to compute the divisors of 2698?

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 2698 by each of the numbers from 1 to 2698 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)

2698 / 1 = 2698 (the remainder is 0, so 1 is a divisor of 2698)
2698 / 2 = 1349 (the remainder is 0, so 2 is a divisor of 2698)
2698 / 3 = 899.33333333333 (the remainder is 1, so 3 is not a divisor of 2698)
...
2698 / 2697 = 1.0003707823508 (the remainder is 1, so 2697 is not a divisor of 2698)
2698 / 2698 = 1 (the remainder is 0, so 2698 is a divisor of 2698)

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 2698 (i.e. 51.942275652882). 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 $\frac{N}{d} = D$ )

2698 / 1 = 2698 (the remainder is 0, so 1 and 2698 are divisors of 2698)
2698 / 2 = 1349 (the remainder is 0, so 2 and 1349 are divisors of 2698)
2698 / 3 = 899.33333333333 (the remainder is 1, so 3 is not a divisor of 2698)
...
2698 / 50 = 53.96 (the remainder is 48, so 50 is not a divisor of 2698)
2698 / 51 = 52.901960784314 (the remainder is 46, so 51 is not a divisor of 2698)