What are the divisors of 4031?

1, 29, 139, 4031

There is a total of 4 positive divisors.
The sum of these divisors is 4200.
The arithmetic mean is 1050.

4 odd divisors

1, 29, 139, 4031

How to compute the divisors of 4031?

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 4031 by each of the numbers from 1 to 4031 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)

4031 / 1 = 4031 (the remainder is 0, so 1 is a divisor of 4031)
4031 / 2 = 2015.5 (the remainder is 1, so 2 is not a divisor of 4031)
4031 / 3 = 1343.6666666667 (the remainder is 2, so 3 is not a divisor of 4031)
...
4031 / 4030 = 1.0002481389578 (the remainder is 1, so 4030 is not a divisor of 4031)
4031 / 4031 = 1 (the remainder is 0, so 4031 is a divisor of 4031)

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 4031 (i.e. 63.4901567174). 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 $\frac{N}{d} = D$ )

4031 / 1 = 4031 (the remainder is 0, so 1 and 4031 are divisors of 4031)
4031 / 2 = 2015.5 (the remainder is 1, so 2 is not a divisor of 4031)
4031 / 3 = 1343.6666666667 (the remainder is 2, so 3 is not a divisor of 4031)
...
4031 / 62 = 65.016129032258 (the remainder is 1, so 62 is not a divisor of 4031)
4031 / 63 = 63.984126984127 (the remainder is 62, so 63 is not a divisor of 4031)