What are the divisors of 168?

1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168

There is a total of 16 positive divisors.
The sum of these divisors is 480.
The arithmetic mean is 30.

12 even divisors

2, 4, 6, 8, 12, 14, 24, 28, 42, 56, 84, 168

4 odd divisors

1, 3, 7, 21

How to compute the divisors of 168?

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 168 by each of the numbers from 1 to 168 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)

168 / 1 = 168 (the remainder is 0, so 1 is a divisor of 168)
168 / 2 = 84 (the remainder is 0, so 2 is a divisor of 168)
168 / 3 = 56 (the remainder is 0, so 3 is a divisor of 168)
...
168 / 167 = 1.0059880239521 (the remainder is 1, so 167 is not a divisor of 168)
168 / 168 = 1 (the remainder is 0, so 168 is a divisor of 168)

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 168 (i.e. 12.961481396816). 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 $\frac{N}{d} = D$ )

168 / 1 = 168 (the remainder is 0, so 1 and 168 are divisors of 168)
168 / 2 = 84 (the remainder is 0, so 2 and 84 are divisors of 168)
168 / 3 = 56 (the remainder is 0, so 3 and 56 are divisors of 168)
...
168 / 11 = 15.272727272727 (the remainder is 3, so 11 is not a divisor of 168)
168 / 12 = 14 (the remainder is 0, so 12 and 14 are divisors of 168)