What are the divisors of 2052?

1, 2, 3, 4, 6, 9, 12, 18, 19, 27, 36, 38, 54, 57, 76, 108, 114, 171, 228, 342, 513, 684, 1026, 2052

There is a total of 24 positive divisors.
The sum of these divisors is 5600.
The arithmetic mean is 233.33333333333.

16 even divisors

2, 4, 6, 12, 18, 36, 38, 54, 76, 108, 114, 228, 342, 684, 1026, 2052

8 odd divisors

1, 3, 9, 19, 27, 57, 171, 513

How to compute the divisors of 2052?

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 2052 by each of the numbers from 1 to 2052 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)

2052 / 1 = 2052 (the remainder is 0, so 1 is a divisor of 2052)
2052 / 2 = 1026 (the remainder is 0, so 2 is a divisor of 2052)
2052 / 3 = 684 (the remainder is 0, so 3 is a divisor of 2052)
...
2052 / 2051 = 1.0004875670405 (the remainder is 1, so 2051 is not a divisor of 2052)
2052 / 2052 = 1 (the remainder is 0, so 2052 is a divisor of 2052)

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 2052 (i.e. 45.299006611624). 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 $\frac{N}{d} = D$ )

2052 / 1 = 2052 (the remainder is 0, so 1 and 2052 are divisors of 2052)
2052 / 2 = 1026 (the remainder is 0, so 2 and 1026 are divisors of 2052)
2052 / 3 = 684 (the remainder is 0, so 3 and 684 are divisors of 2052)
...
2052 / 44 = 46.636363636364 (the remainder is 28, so 44 is not a divisor of 2052)
2052 / 45 = 45.6 (the remainder is 27, so 45 is not a divisor of 2052)