What are the divisors of 123?

1, 3, 41, 123

There is a total of 4 positive divisors.
The sum of these divisors is 168.
The arithmetic mean is 42.

4 odd divisors

1, 3, 41, 123

How to compute the divisors of 123?

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 123 by each of the numbers from 1 to 123 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)

123 / 1 = 123 (the remainder is 0, so 1 is a divisor of 123)
123 / 2 = 61.5 (the remainder is 1, so 2 is not a divisor of 123)
123 / 3 = 41 (the remainder is 0, so 3 is a divisor of 123)
...
123 / 122 = 1.0081967213115 (the remainder is 1, so 122 is not a divisor of 123)
123 / 123 = 1 (the remainder is 0, so 123 is a divisor of 123)

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 123 (i.e. 11.090536506409). 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 $\frac{N}{d} = D$ )

123 / 1 = 123 (the remainder is 0, so 1 and 123 are divisors of 123)
123 / 2 = 61.5 (the remainder is 1, so 2 is not a divisor of 123)
123 / 3 = 41 (the remainder is 0, so 3 and 41 are divisors of 123)
...
123 / 10 = 12.3 (the remainder is 3, so 10 is not a divisor of 123)
123 / 11 = 11.181818181818 (the remainder is 2, so 11 is not a divisor of 123)