What are the divisors of 392?

1, 2, 4, 7, 8, 14, 28, 49, 56, 98, 196, 392

There is a total of 12 positive divisors.
The sum of these divisors is 855.
The arithmetic mean is 71.25.

9 even divisors

2, 4, 8, 14, 28, 56, 98, 196, 392

3 odd divisors

1, 7, 49

How to compute the divisors of 392?

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 392 by each of the numbers from 1 to 392 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)

392 / 1 = 392 (the remainder is 0, so 1 is a divisor of 392)
392 / 2 = 196 (the remainder is 0, so 2 is a divisor of 392)
392 / 3 = 130.66666666667 (the remainder is 2, so 3 is not a divisor of 392)
...
392 / 391 = 1.002557544757 (the remainder is 1, so 391 is not a divisor of 392)
392 / 392 = 1 (the remainder is 0, so 392 is a divisor of 392)

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 392 (i.e. 19.798989873223). 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 $\frac{N}{d} = D$ )

392 / 1 = 392 (the remainder is 0, so 1 and 392 are divisors of 392)
392 / 2 = 196 (the remainder is 0, so 2 and 196 are divisors of 392)
392 / 3 = 130.66666666667 (the remainder is 2, so 3 is not a divisor of 392)
...
392 / 18 = 21.777777777778 (the remainder is 14, so 18 is not a divisor of 392)
392 / 19 = 20.631578947368 (the remainder is 12, so 19 is not a divisor of 392)