What are the divisors of 107?

1, 107

There is a total of 2 positive divisors.
The sum of these divisors is 108.
The arithmetic mean is 54.

2 odd divisors

1, 107

How to compute the divisors of 107?

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 107 by each of the numbers from 1 to 107 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)

107 / 1 = 107 (the remainder is 0, so 1 is a divisor of 107)
107 / 2 = 53.5 (the remainder is 1, so 2 is not a divisor of 107)
107 / 3 = 35.666666666667 (the remainder is 2, so 3 is not a divisor of 107)
...
107 / 106 = 1.0094339622642 (the remainder is 1, so 106 is not a divisor of 107)
107 / 107 = 1 (the remainder is 0, so 107 is a divisor of 107)

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 107 (i.e. 10.344080432789). 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 $\frac{N}{d} = D$ )

107 / 1 = 107 (the remainder is 0, so 1 and 107 are divisors of 107)
107 / 2 = 53.5 (the remainder is 1, so 2 is not a divisor of 107)
107 / 3 = 35.666666666667 (the remainder is 2, so 3 is not a divisor of 107)
...
107 / 9 = 11.888888888889 (the remainder is 8, so 9 is not a divisor of 107)
107 / 10 = 10.7 (the remainder is 7, so 10 is not a divisor of 107)