What are the divisors of 131?

1, 131

There is a total of 2 positive divisors.
The sum of these divisors is 132.
The arithmetic mean is 66.

2 odd divisors

1, 131

How to compute the divisors of 131?

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 131 by each of the numbers from 1 to 131 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)

131 / 1 = 131 (the remainder is 0, so 1 is a divisor of 131)
131 / 2 = 65.5 (the remainder is 1, so 2 is not a divisor of 131)
131 / 3 = 43.666666666667 (the remainder is 2, so 3 is not a divisor of 131)
...
131 / 130 = 1.0076923076923 (the remainder is 1, so 130 is not a divisor of 131)
131 / 131 = 1 (the remainder is 0, so 131 is a divisor of 131)

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 131 (i.e. 11.44552314226). 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 $\frac{N}{d} = D$ )

131 / 1 = 131 (the remainder is 0, so 1 and 131 are divisors of 131)
131 / 2 = 65.5 (the remainder is 1, so 2 is not a divisor of 131)
131 / 3 = 43.666666666667 (the remainder is 2, so 3 is not a divisor of 131)
...
131 / 10 = 13.1 (the remainder is 1, so 10 is not a divisor of 131)
131 / 11 = 11.909090909091 (the remainder is 10, so 11 is not a divisor of 131)