What are the divisors of 165?

1, 3, 5, 11, 15, 33, 55, 165

There is a total of 8 positive divisors.
The sum of these divisors is 288.
The arithmetic mean is 36.

8 odd divisors

1, 3, 5, 11, 15, 33, 55, 165

How to compute the divisors of 165?

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 165 by each of the numbers from 1 to 165 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)

165 / 1 = 165 (the remainder is 0, so 1 is a divisor of 165)
165 / 2 = 82.5 (the remainder is 1, so 2 is not a divisor of 165)
165 / 3 = 55 (the remainder is 0, so 3 is a divisor of 165)
...
165 / 164 = 1.0060975609756 (the remainder is 1, so 164 is not a divisor of 165)
165 / 165 = 1 (the remainder is 0, so 165 is a divisor of 165)

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 165 (i.e. 12.845232578665). 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 $\frac{N}{d} = D$ )

165 / 1 = 165 (the remainder is 0, so 1 and 165 are divisors of 165)
165 / 2 = 82.5 (the remainder is 1, so 2 is not a divisor of 165)
165 / 3 = 55 (the remainder is 0, so 3 and 55 are divisors of 165)
...
165 / 11 = 15 (the remainder is 0, so 11 and 15 are divisors of 165)
165 / 12 = 13.75 (the remainder is 9, so 12 is not a divisor of 165)