What are the divisors of 130?

1, 2, 5, 10, 13, 26, 65, 130

There is a total of 8 positive divisors.
The sum of these divisors is 252.
The arithmetic mean is 31.5.

4 even divisors

2, 10, 26, 130

4 odd divisors

1, 5, 13, 65

How to compute the divisors of 130?

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 130 by each of the numbers from 1 to 130 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)

130 / 1 = 130 (the remainder is 0, so 1 is a divisor of 130)
130 / 2 = 65 (the remainder is 0, so 2 is a divisor of 130)
130 / 3 = 43.333333333333 (the remainder is 1, so 3 is not a divisor of 130)
...
130 / 129 = 1.0077519379845 (the remainder is 1, so 129 is not a divisor of 130)
130 / 130 = 1 (the remainder is 0, so 130 is a divisor of 130)

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 130 (i.e. 11.401754250991). 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 $\frac{N}{d} = D$ )

130 / 1 = 130 (the remainder is 0, so 1 and 130 are divisors of 130)
130 / 2 = 65 (the remainder is 0, so 2 and 65 are divisors of 130)
130 / 3 = 43.333333333333 (the remainder is 1, so 3 is not a divisor of 130)
...
130 / 10 = 13 (the remainder is 0, so 10 and 13 are divisors of 130)
130 / 11 = 11.818181818182 (the remainder is 9, so 11 is not a divisor of 130)