What are the divisors of 111?

1, 3, 37, 111

There is a total of 4 positive divisors.
The sum of these divisors is 152.
The arithmetic mean is 38.

4 odd divisors

1, 3, 37, 111

How to compute the divisors of 111?

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 111 by each of the numbers from 1 to 111 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)

111 / 1 = 111 (the remainder is 0, so 1 is a divisor of 111)
111 / 2 = 55.5 (the remainder is 1, so 2 is not a divisor of 111)
111 / 3 = 37 (the remainder is 0, so 3 is a divisor of 111)
...
111 / 110 = 1.0090909090909 (the remainder is 1, so 110 is not a divisor of 111)
111 / 111 = 1 (the remainder is 0, so 111 is a divisor of 111)

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 111 (i.e. 10.535653752853). 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 $\frac{N}{d} = D$ )

111 / 1 = 111 (the remainder is 0, so 1 and 111 are divisors of 111)
111 / 2 = 55.5 (the remainder is 1, so 2 is not a divisor of 111)
111 / 3 = 37 (the remainder is 0, so 3 and 37 are divisors of 111)
...
111 / 9 = 12.333333333333 (the remainder is 3, so 9 is not a divisor of 111)
111 / 10 = 11.1 (the remainder is 1, so 10 is not a divisor of 111)