What are the divisors of 2196?

1, 2, 3, 4, 6, 9, 12, 18, 36, 61, 122, 183, 244, 366, 549, 732, 1098, 2196

There is a total of 18 positive divisors.
The sum of these divisors is 5642.
The arithmetic mean is 313.44444444444.

12 even divisors

2, 4, 6, 12, 18, 36, 122, 244, 366, 732, 1098, 2196

6 odd divisors

1, 3, 9, 61, 183, 549

How to compute the divisors of 2196?

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 2196 by each of the numbers from 1 to 2196 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)

2196 / 1 = 2196 (the remainder is 0, so 1 is a divisor of 2196)
2196 / 2 = 1098 (the remainder is 0, so 2 is a divisor of 2196)
2196 / 3 = 732 (the remainder is 0, so 3 is a divisor of 2196)
...
2196 / 2195 = 1.0004555808656 (the remainder is 1, so 2195 is not a divisor of 2196)
2196 / 2196 = 1 (the remainder is 0, so 2196 is a divisor of 2196)

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 2196 (i.e. 46.86149805544). 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 $\frac{N}{d} = D$ )

2196 / 1 = 2196 (the remainder is 0, so 1 and 2196 are divisors of 2196)
2196 / 2 = 1098 (the remainder is 0, so 2 and 1098 are divisors of 2196)
2196 / 3 = 732 (the remainder is 0, so 3 and 732 are divisors of 2196)
...
2196 / 45 = 48.8 (the remainder is 36, so 45 is not a divisor of 2196)
2196 / 46 = 47.739130434783 (the remainder is 34, so 46 is not a divisor of 2196)