What are the divisors of 166?

1, 2, 83, 166

There is a total of 4 positive divisors.
The sum of these divisors is 252.
The arithmetic mean is 63.

2 even divisors

2, 166

2 odd divisors

1, 83

How to compute the divisors of 166?

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 166 by each of the numbers from 1 to 166 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)

166 / 1 = 166 (the remainder is 0, so 1 is a divisor of 166)
166 / 2 = 83 (the remainder is 0, so 2 is a divisor of 166)
166 / 3 = 55.333333333333 (the remainder is 1, so 3 is not a divisor of 166)
...
166 / 165 = 1.0060606060606 (the remainder is 1, so 165 is not a divisor of 166)
166 / 166 = 1 (the remainder is 0, so 166 is a divisor of 166)

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 166 (i.e. 12.884098726725). 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 $\frac{N}{d} = D$ )

166 / 1 = 166 (the remainder is 0, so 1 and 166 are divisors of 166)
166 / 2 = 83 (the remainder is 0, so 2 and 83 are divisors of 166)
166 / 3 = 55.333333333333 (the remainder is 1, so 3 is not a divisor of 166)
...
166 / 11 = 15.090909090909 (the remainder is 1, so 11 is not a divisor of 166)
166 / 12 = 13.833333333333 (the remainder is 10, so 12 is not a divisor of 166)