What are the divisors of 109?

1, 109

There is a total of 2 positive divisors.
The sum of these divisors is 110.
The arithmetic mean is 55.

2 odd divisors

1, 109

How to compute the divisors of 109?

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 109 by each of the numbers from 1 to 109 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)

109 / 1 = 109 (the remainder is 0, so 1 is a divisor of 109)
109 / 2 = 54.5 (the remainder is 1, so 2 is not a divisor of 109)
109 / 3 = 36.333333333333 (the remainder is 1, so 3 is not a divisor of 109)
...
109 / 108 = 1.0092592592593 (the remainder is 1, so 108 is not a divisor of 109)
109 / 109 = 1 (the remainder is 0, so 109 is a divisor of 109)

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 109 (i.e. 10.440306508911). 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 $\frac{N}{d} = D$ )

109 / 1 = 109 (the remainder is 0, so 1 and 109 are divisors of 109)
109 / 2 = 54.5 (the remainder is 1, so 2 is not a divisor of 109)
109 / 3 = 36.333333333333 (the remainder is 1, so 3 is not a divisor of 109)
...
109 / 9 = 12.111111111111 (the remainder is 1, so 9 is not a divisor of 109)
109 / 10 = 10.9 (the remainder is 9, so 10 is not a divisor of 109)