What are the divisors of 1260?

1, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 15, 18, 20, 21, 28, 30, 35, 36, 42, 45, 60, 63, 70, 84, 90, 105, 126, 140, 180, 210, 252, 315, 420, 630, 1260

There is a total of 36 positive divisors.
The sum of these divisors is 4368.
The arithmetic mean is 121.33333333333.

24 even divisors

2, 4, 6, 10, 12, 14, 18, 20, 28, 30, 36, 42, 60, 70, 84, 90, 126, 140, 180, 210, 252, 420, 630, 1260

12 odd divisors

1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315

How to compute the divisors of 1260?

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 1260 by each of the numbers from 1 to 1260 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)

1260 / 1 = 1260 (the remainder is 0, so 1 is a divisor of 1260)
1260 / 2 = 630 (the remainder is 0, so 2 is a divisor of 1260)
1260 / 3 = 420 (the remainder is 0, so 3 is a divisor of 1260)
...
1260 / 1259 = 1.0007942811755 (the remainder is 1, so 1259 is not a divisor of 1260)
1260 / 1260 = 1 (the remainder is 0, so 1260 is a divisor of 1260)

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 1260 (i.e. 35.496478698598). 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 $\frac{N}{d} = D$ )

1260 / 1 = 1260 (the remainder is 0, so 1 and 1260 are divisors of 1260)
1260 / 2 = 630 (the remainder is 0, so 2 and 630 are divisors of 1260)
1260 / 3 = 420 (the remainder is 0, so 3 and 420 are divisors of 1260)
...
1260 / 34 = 37.058823529412 (the remainder is 2, so 34 is not a divisor of 1260)
1260 / 35 = 36 (the remainder is 0, so 35 and 36 are divisors of 1260)