What are the divisors of 99?

1, 3, 9, 11, 33, 99

There is a total of 6 positive divisors.
The sum of these divisors is 156.
The arithmetic mean is 26.

6 odd divisors

1, 3, 9, 11, 33, 99

How to compute the divisors of 99?

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 99 by each of the numbers from 1 to 99 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)

99 / 1 = 99 (the remainder is 0, so 1 is a divisor of 99)
99 / 2 = 49.5 (the remainder is 1, so 2 is not a divisor of 99)
99 / 3 = 33 (the remainder is 0, so 3 is a divisor of 99)
...
99 / 98 = 1.0102040816327 (the remainder is 1, so 98 is not a divisor of 99)
99 / 99 = 1 (the remainder is 0, so 99 is a divisor of 99)

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 99 (i.e. 9.9498743710662). 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 $\frac{N}{d} = D$ )

99 / 1 = 99 (the remainder is 0, so 1 and 99 are divisors of 99)
99 / 2 = 49.5 (the remainder is 1, so 2 is not a divisor of 99)
99 / 3 = 33 (the remainder is 0, so 3 and 33 are divisors of 99)
...
99 / 8 = 12.375 (the remainder is 3, so 8 is not a divisor of 99)
99 / 9 = 11 (the remainder is 0, so 9 and 11 are divisors of 99)