What are the divisors of 2697?

1, 3, 29, 31, 87, 93, 899, 2697

There is a total of 8 positive divisors.
The sum of these divisors is 3840.
The arithmetic mean is 480.

8 odd divisors

1, 3, 29, 31, 87, 93, 899, 2697

How to compute the divisors of 2697?

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 2697 by each of the numbers from 1 to 2697 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)

2697 / 1 = 2697 (the remainder is 0, so 1 is a divisor of 2697)
2697 / 2 = 1348.5 (the remainder is 1, so 2 is not a divisor of 2697)
2697 / 3 = 899 (the remainder is 0, so 3 is a divisor of 2697)
...
2697 / 2696 = 1.0003709198813 (the remainder is 1, so 2696 is not a divisor of 2697)
2697 / 2697 = 1 (the remainder is 0, so 2697 is a divisor of 2697)

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 2697 (i.e. 51.932648690395). 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 $\frac{N}{d} = D$ )

2697 / 1 = 2697 (the remainder is 0, so 1 and 2697 are divisors of 2697)
2697 / 2 = 1348.5 (the remainder is 1, so 2 is not a divisor of 2697)
2697 / 3 = 899 (the remainder is 0, so 3 and 899 are divisors of 2697)
...
2697 / 50 = 53.94 (the remainder is 47, so 50 is not a divisor of 2697)
2697 / 51 = 52.882352941176 (the remainder is 45, so 51 is not a divisor of 2697)