What are the divisors of 672?

1, 2, 3, 4, 6, 7, 8, 12, 14, 16, 21, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672

There is a total of 24 positive divisors.
The sum of these divisors is 2016.
The arithmetic mean is 84.

20 even divisors

2, 4, 6, 8, 12, 14, 16, 24, 28, 32, 42, 48, 56, 84, 96, 112, 168, 224, 336, 672

4 odd divisors

1, 3, 7, 21

How to compute the divisors of 672?

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 672 by each of the numbers from 1 to 672 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)

672 / 1 = 672 (the remainder is 0, so 1 is a divisor of 672)
672 / 2 = 336 (the remainder is 0, so 2 is a divisor of 672)
672 / 3 = 224 (the remainder is 0, so 3 is a divisor of 672)
...
672 / 671 = 1.0014903129657 (the remainder is 1, so 671 is not a divisor of 672)
672 / 672 = 1 (the remainder is 0, so 672 is a divisor of 672)

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 672 (i.e. 25.922962793631). 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 $\frac{N}{d} = D$ )

672 / 1 = 672 (the remainder is 0, so 1 and 672 are divisors of 672)
672 / 2 = 336 (the remainder is 0, so 2 and 336 are divisors of 672)
672 / 3 = 224 (the remainder is 0, so 3 and 224 are divisors of 672)
...
672 / 24 = 28 (the remainder is 0, so 24 and 28 are divisors of 672)
672 / 25 = 26.88 (the remainder is 22, so 25 is not a divisor of 672)