What are the divisors of 110?

1, 2, 5, 10, 11, 22, 55, 110

There is a total of 8 positive divisors.
The sum of these divisors is 216.
The arithmetic mean is 27.

4 even divisors

2, 10, 22, 110

4 odd divisors

1, 5, 11, 55

How to compute the divisors of 110?

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 110 by each of the numbers from 1 to 110 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)

110 / 1 = 110 (the remainder is 0, so 1 is a divisor of 110)
110 / 2 = 55 (the remainder is 0, so 2 is a divisor of 110)
110 / 3 = 36.666666666667 (the remainder is 2, so 3 is not a divisor of 110)
...
110 / 109 = 1.0091743119266 (the remainder is 1, so 109 is not a divisor of 110)
110 / 110 = 1 (the remainder is 0, so 110 is a divisor of 110)

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 110 (i.e. 10.488088481702). 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 $\frac{N}{d} = D$ )

110 / 1 = 110 (the remainder is 0, so 1 and 110 are divisors of 110)
110 / 2 = 55 (the remainder is 0, so 2 and 55 are divisors of 110)
110 / 3 = 36.666666666667 (the remainder is 2, so 3 is not a divisor of 110)
...
110 / 9 = 12.222222222222 (the remainder is 2, so 9 is not a divisor of 110)
110 / 10 = 11 (the remainder is 0, so 10 and 11 are divisors of 110)