What are the divisors of 84?

1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84

There is a total of 12 positive divisors.
The sum of these divisors is 224.
The arithmetic mean is 18.666666666667.

8 even divisors

2, 4, 6, 12, 14, 28, 42, 84

4 odd divisors

1, 3, 7, 21

How to compute the divisors of 84?

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 84 by each of the numbers from 1 to 84 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)

84 / 1 = 84 (the remainder is 0, so 1 is a divisor of 84)
84 / 2 = 42 (the remainder is 0, so 2 is a divisor of 84)
84 / 3 = 28 (the remainder is 0, so 3 is a divisor of 84)
...
84 / 83 = 1.0120481927711 (the remainder is 1, so 83 is not a divisor of 84)
84 / 84 = 1 (the remainder is 0, so 84 is a divisor of 84)

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 84 (i.e. 9.1651513899117). 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 $\frac{N}{d} = D$ )

84 / 1 = 84 (the remainder is 0, so 1 and 84 are divisors of 84)
84 / 2 = 42 (the remainder is 0, so 2 and 42 are divisors of 84)
84 / 3 = 28 (the remainder is 0, so 3 and 28 are divisors of 84)
...
84 / 8 = 10.5 (the remainder is 4, so 8 is not a divisor of 84)
84 / 9 = 9.3333333333333 (the remainder is 3, so 9 is not a divisor of 84)