What are the divisors of 448?

1, 2, 4, 7, 8, 14, 16, 28, 32, 56, 64, 112, 224, 448

There is a total of 14 positive divisors.
The sum of these divisors is 1016.
The arithmetic mean is 72.571428571429.

12 even divisors

2, 4, 8, 14, 16, 28, 32, 56, 64, 112, 224, 448

2 odd divisors

1, 7

How to compute the divisors of 448?

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 448 by each of the numbers from 1 to 448 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)

448 / 1 = 448 (the remainder is 0, so 1 is a divisor of 448)
448 / 2 = 224 (the remainder is 0, so 2 is a divisor of 448)
448 / 3 = 149.33333333333 (the remainder is 1, so 3 is not a divisor of 448)
...
448 / 447 = 1.0022371364653 (the remainder is 1, so 447 is not a divisor of 448)
448 / 448 = 1 (the remainder is 0, so 448 is a divisor of 448)

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 448 (i.e. 21.166010488517). 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 $\frac{N}{d} = D$ )

448 / 1 = 448 (the remainder is 0, so 1 and 448 are divisors of 448)
448 / 2 = 224 (the remainder is 0, so 2 and 224 are divisors of 448)
448 / 3 = 149.33333333333 (the remainder is 1, so 3 is not a divisor of 448)
...
448 / 20 = 22.4 (the remainder is 8, so 20 is not a divisor of 448)
448 / 21 = 21.333333333333 (the remainder is 7, so 21 is not a divisor of 448)