What are the divisors of 1380?

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 23, 30, 46, 60, 69, 92, 115, 138, 230, 276, 345, 460, 690, 1380

There is a total of 24 positive divisors.
The sum of these divisors is 4032.
The arithmetic mean is 168.

16 even divisors

2, 4, 6, 10, 12, 20, 30, 46, 60, 92, 138, 230, 276, 460, 690, 1380

8 odd divisors

1, 3, 5, 15, 23, 69, 115, 345

How to compute the divisors of 1380?

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 1380 by each of the numbers from 1 to 1380 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)

1380 / 1 = 1380 (the remainder is 0, so 1 is a divisor of 1380)
1380 / 2 = 690 (the remainder is 0, so 2 is a divisor of 1380)
1380 / 3 = 460 (the remainder is 0, so 3 is a divisor of 1380)
...
1380 / 1379 = 1.0007251631617 (the remainder is 1, so 1379 is not a divisor of 1380)
1380 / 1380 = 1 (the remainder is 0, so 1380 is a divisor of 1380)

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 1380 (i.e. 37.148351242013). 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 $\frac{N}{d} = D$ )

1380 / 1 = 1380 (the remainder is 0, so 1 and 1380 are divisors of 1380)
1380 / 2 = 690 (the remainder is 0, so 2 and 690 are divisors of 1380)
1380 / 3 = 460 (the remainder is 0, so 3 and 460 are divisors of 1380)
...
1380 / 36 = 38.333333333333 (the remainder is 12, so 36 is not a divisor of 1380)
1380 / 37 = 37.297297297297 (the remainder is 11, so 37 is not a divisor of 1380)