What are the divisors of 8142?

1, 2, 3, 6, 23, 46, 59, 69, 118, 138, 177, 354, 1357, 2714, 4071, 8142

There is a total of 16 positive divisors.
The sum of these divisors is 17280.
The arithmetic mean is 1080.

8 even divisors

2, 6, 46, 118, 138, 354, 2714, 8142

8 odd divisors

1, 3, 23, 59, 69, 177, 1357, 4071

How to compute the divisors of 8142?

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 8142 by each of the numbers from 1 to 8142 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)

8142 / 1 = 8142 (the remainder is 0, so 1 is a divisor of 8142)
8142 / 2 = 4071 (the remainder is 0, so 2 is a divisor of 8142)
8142 / 3 = 2714 (the remainder is 0, so 3 is a divisor of 8142)
...
8142 / 8141 = 1.0001228350326 (the remainder is 1, so 8141 is not a divisor of 8142)
8142 / 8142 = 1 (the remainder is 0, so 8142 is a divisor of 8142)

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 8142 (i.e. 90.233031645845). 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 $\frac{N}{d} = D$ )

8142 / 1 = 8142 (the remainder is 0, so 1 and 8142 are divisors of 8142)
8142 / 2 = 4071 (the remainder is 0, so 2 and 4071 are divisors of 8142)
8142 / 3 = 2714 (the remainder is 0, so 3 and 2714 are divisors of 8142)
...
8142 / 89 = 91.483146067416 (the remainder is 43, so 89 is not a divisor of 8142)
8142 / 90 = 90.466666666667 (the remainder is 42, so 90 is not a divisor of 8142)