What are the divisors of 224?

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

There is a total of 12 positive divisors.
The sum of these divisors is 504.
The arithmetic mean is 42.

10 even divisors

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

2 odd divisors

1, 7

How to compute the divisors of 224?

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 224 by each of the numbers from 1 to 224 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)

224 / 1 = 224 (the remainder is 0, so 1 is a divisor of 224)
224 / 2 = 112 (the remainder is 0, so 2 is a divisor of 224)
224 / 3 = 74.666666666667 (the remainder is 2, so 3 is not a divisor of 224)
...
224 / 223 = 1.0044843049327 (the remainder is 1, so 223 is not a divisor of 224)
224 / 224 = 1 (the remainder is 0, so 224 is a divisor of 224)

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 224 (i.e. 14.966629547096). 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 $\frac{N}{d} = D$ )

224 / 1 = 224 (the remainder is 0, so 1 and 224 are divisors of 224)
224 / 2 = 112 (the remainder is 0, so 2 and 112 are divisors of 224)
224 / 3 = 74.666666666667 (the remainder is 2, so 3 is not a divisor of 224)
...
224 / 13 = 17.230769230769 (the remainder is 3, so 13 is not a divisor of 224)
224 / 14 = 16 (the remainder is 0, so 14 and 16 are divisors of 224)