What are the divisors of 171?

1, 3, 9, 19, 57, 171

There is a total of 6 positive divisors.
The sum of these divisors is 260.
The arithmetic mean is 43.333333333333.

6 odd divisors

1, 3, 9, 19, 57, 171

How to compute the divisors of 171?

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 171 by each of the numbers from 1 to 171 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)

171 / 1 = 171 (the remainder is 0, so 1 is a divisor of 171)
171 / 2 = 85.5 (the remainder is 1, so 2 is not a divisor of 171)
171 / 3 = 57 (the remainder is 0, so 3 is a divisor of 171)
...
171 / 170 = 1.0058823529412 (the remainder is 1, so 170 is not a divisor of 171)
171 / 171 = 1 (the remainder is 0, so 171 is a divisor of 171)

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 171 (i.e. 13.076696830622). 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 $\frac{N}{d} = D$ )

171 / 1 = 171 (the remainder is 0, so 1 and 171 are divisors of 171)
171 / 2 = 85.5 (the remainder is 1, so 2 is not a divisor of 171)
171 / 3 = 57 (the remainder is 0, so 3 and 57 are divisors of 171)
...
171 / 12 = 14.25 (the remainder is 3, so 12 is not a divisor of 171)
171 / 13 = 13.153846153846 (the remainder is 2, so 13 is not a divisor of 171)