What are the divisors of 167?

1, 167

There is a total of 2 positive divisors.
The sum of these divisors is 168.
The arithmetic mean is 84.

2 odd divisors

1, 167

How to compute the divisors of 167?

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 167 by each of the numbers from 1 to 167 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)

167 / 1 = 167 (the remainder is 0, so 1 is a divisor of 167)
167 / 2 = 83.5 (the remainder is 1, so 2 is not a divisor of 167)
167 / 3 = 55.666666666667 (the remainder is 2, so 3 is not a divisor of 167)
...
167 / 166 = 1.0060240963855 (the remainder is 1, so 166 is not a divisor of 167)
167 / 167 = 1 (the remainder is 0, so 167 is a divisor of 167)

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 167 (i.e. 12.92284798332). 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 $\frac{N}{d} = D$ )

167 / 1 = 167 (the remainder is 0, so 1 and 167 are divisors of 167)
167 / 2 = 83.5 (the remainder is 1, so 2 is not a divisor of 167)
167 / 3 = 55.666666666667 (the remainder is 2, so 3 is not a divisor of 167)
...
167 / 11 = 15.181818181818 (the remainder is 2, so 11 is not a divisor of 167)
167 / 12 = 13.916666666667 (the remainder is 11, so 12 is not a divisor of 167)