What are the divisors of 3780?

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 27, 28, 30, 35, 36, 42, 45, 54, 60, 63, 70, 84, 90, 105, 108, 126, 135, 140, 180, 189, 210, 252, 270, 315, 378, 420, 540, 630, 756, 945, 1260, 1890, 3780

There is a total of 48 positive divisors.
The sum of these divisors is 13440.
The arithmetic mean is 280.

32 even divisors

2, 4, 6, 10, 12, 14, 18, 20, 28, 30, 36, 42, 54, 60, 70, 84, 90, 108, 126, 140, 180, 210, 252, 270, 378, 420, 540, 630, 756, 1260, 1890, 3780

16 odd divisors

1, 3, 5, 7, 9, 15, 21, 27, 35, 45, 63, 105, 135, 189, 315, 945

How to compute the divisors of 3780?

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 3780 by each of the numbers from 1 to 3780 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)

3780 / 1 = 3780 (the remainder is 0, so 1 is a divisor of 3780)
3780 / 2 = 1890 (the remainder is 0, so 2 is a divisor of 3780)
3780 / 3 = 1260 (the remainder is 0, so 3 is a divisor of 3780)
...
3780 / 3779 = 1.0002646202699 (the remainder is 1, so 3779 is not a divisor of 3780)
3780 / 3780 = 1 (the remainder is 0, so 3780 is a divisor of 3780)

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 3780 (i.e. 61.481704595758). 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 $\frac{N}{d} = D$ )

3780 / 1 = 3780 (the remainder is 0, so 1 and 3780 are divisors of 3780)
3780 / 2 = 1890 (the remainder is 0, so 2 and 1890 are divisors of 3780)
3780 / 3 = 1260 (the remainder is 0, so 3 and 1260 are divisors of 3780)
...
3780 / 60 = 63 (the remainder is 0, so 60 and 63 are divisors of 3780)
3780 / 61 = 61.967213114754 (the remainder is 59, so 61 is not a divisor of 3780)