What are the divisors of 192?

1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192

There is a total of 14 positive divisors.
The sum of these divisors is 508.
The arithmetic mean is 36.285714285714.

12 even divisors

2, 4, 6, 8, 12, 16, 24, 32, 48, 64, 96, 192

2 odd divisors

1, 3

How to compute the divisors of 192?

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 192 by each of the numbers from 1 to 192 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)

192 / 1 = 192 (the remainder is 0, so 1 is a divisor of 192)
192 / 2 = 96 (the remainder is 0, so 2 is a divisor of 192)
192 / 3 = 64 (the remainder is 0, so 3 is a divisor of 192)
...
192 / 191 = 1.0052356020942 (the remainder is 1, so 191 is not a divisor of 192)
192 / 192 = 1 (the remainder is 0, so 192 is a divisor of 192)

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 192 (i.e. 13.856406460551). 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 $\frac{N}{d} = D$ )

192 / 1 = 192 (the remainder is 0, so 1 and 192 are divisors of 192)
192 / 2 = 96 (the remainder is 0, so 2 and 96 are divisors of 192)
192 / 3 = 64 (the remainder is 0, so 3 and 64 are divisors of 192)
...
192 / 12 = 16 (the remainder is 0, so 12 and 16 are divisors of 192)
192 / 13 = 14.769230769231 (the remainder is 10, so 13 is not a divisor of 192)