What are the divisors of 1780?

1, 2, 4, 5, 10, 20, 89, 178, 356, 445, 890, 1780

There is a total of 12 positive divisors.
The sum of these divisors is 3780.
The arithmetic mean is 315.

8 even divisors

2, 4, 10, 20, 178, 356, 890, 1780

4 odd divisors

1, 5, 89, 445

How to compute the divisors of 1780?

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 1780 by each of the numbers from 1 to 1780 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)

1780 / 1 = 1780 (the remainder is 0, so 1 is a divisor of 1780)
1780 / 2 = 890 (the remainder is 0, so 2 is a divisor of 1780)
1780 / 3 = 593.33333333333 (the remainder is 1, so 3 is not a divisor of 1780)
...
1780 / 1779 = 1.0005621135469 (the remainder is 1, so 1779 is not a divisor of 1780)
1780 / 1780 = 1 (the remainder is 0, so 1780 is a divisor of 1780)

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 1780 (i.e. 42.190046219458). 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 $\frac{N}{d} = D$ )

1780 / 1 = 1780 (the remainder is 0, so 1 and 1780 are divisors of 1780)
1780 / 2 = 890 (the remainder is 0, so 2 and 890 are divisors of 1780)
1780 / 3 = 593.33333333333 (the remainder is 1, so 3 is not a divisor of 1780)
...
1780 / 41 = 43.414634146341 (the remainder is 17, so 41 is not a divisor of 1780)
1780 / 42 = 42.380952380952 (the remainder is 16, so 42 is not a divisor of 1780)