What are the divisors of 1300?

1, 2, 4, 5, 10, 13, 20, 25, 26, 50, 52, 65, 100, 130, 260, 325, 650, 1300

There is a total of 18 positive divisors.
The sum of these divisors is 3038.
The arithmetic mean is 168.77777777778.

12 even divisors

2, 4, 10, 20, 26, 50, 52, 100, 130, 260, 650, 1300

6 odd divisors

1, 5, 13, 25, 65, 325

How to compute the divisors of 1300?

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 1300 by each of the numbers from 1 to 1300 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)

1300 / 1 = 1300 (the remainder is 0, so 1 is a divisor of 1300)
1300 / 2 = 650 (the remainder is 0, so 2 is a divisor of 1300)
1300 / 3 = 433.33333333333 (the remainder is 1, so 3 is not a divisor of 1300)
...
1300 / 1299 = 1.0007698229407 (the remainder is 1, so 1299 is not a divisor of 1300)
1300 / 1300 = 1 (the remainder is 0, so 1300 is a divisor of 1300)

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 1300 (i.e. 36.05551275464). 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 $\frac{N}{d} = D$ )

1300 / 1 = 1300 (the remainder is 0, so 1 and 1300 are divisors of 1300)
1300 / 2 = 650 (the remainder is 0, so 2 and 650 are divisors of 1300)
1300 / 3 = 433.33333333333 (the remainder is 1, so 3 is not a divisor of 1300)
...
1300 / 35 = 37.142857142857 (the remainder is 5, so 35 is not a divisor of 1300)
1300 / 36 = 36.111111111111 (the remainder is 4, so 36 is not a divisor of 1300)