What are the divisors of 4142?

1, 2, 19, 38, 109, 218, 2071, 4142

There is a total of 8 positive divisors.
The sum of these divisors is 6600.
The arithmetic mean is 825.

4 even divisors

2, 38, 218, 4142

4 odd divisors

1, 19, 109, 2071

How to compute the divisors of 4142?

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 4142 by each of the numbers from 1 to 4142 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)

4142 / 1 = 4142 (the remainder is 0, so 1 is a divisor of 4142)
4142 / 2 = 2071 (the remainder is 0, so 2 is a divisor of 4142)
4142 / 3 = 1380.6666666667 (the remainder is 2, so 3 is not a divisor of 4142)
...
4142 / 4141 = 1.0002414875634 (the remainder is 1, so 4141 is not a divisor of 4142)
4142 / 4142 = 1 (the remainder is 0, so 4142 is a divisor of 4142)

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 4142 (i.e. 64.358371638816). 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 $\frac{N}{d} = D$ )

4142 / 1 = 4142 (the remainder is 0, so 1 and 4142 are divisors of 4142)
4142 / 2 = 2071 (the remainder is 0, so 2 and 2071 are divisors of 4142)
4142 / 3 = 1380.6666666667 (the remainder is 2, so 3 is not a divisor of 4142)
...
4142 / 63 = 65.746031746032 (the remainder is 47, so 63 is not a divisor of 4142)
4142 / 64 = 64.71875 (the remainder is 46, so 64 is not a divisor of 4142)