What are the divisors of 4111?

1, 4111

There is a total of 2 positive divisors.
The sum of these divisors is 4112.
The arithmetic mean is 2056.

2 odd divisors

1, 4111

How to compute the divisors of 4111?

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 4111 by each of the numbers from 1 to 4111 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)

4111 / 1 = 4111 (the remainder is 0, so 1 is a divisor of 4111)
4111 / 2 = 2055.5 (the remainder is 1, so 2 is not a divisor of 4111)
4111 / 3 = 1370.3333333333 (the remainder is 1, so 3 is not a divisor of 4111)
...
4111 / 4110 = 1.0002433090024 (the remainder is 1, so 4110 is not a divisor of 4111)
4111 / 4111 = 1 (the remainder is 0, so 4111 is a divisor of 4111)

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 4111 (i.e. 64.117080407642). 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 $\frac{N}{d} = D$ )

4111 / 1 = 4111 (the remainder is 0, so 1 and 4111 are divisors of 4111)
4111 / 2 = 2055.5 (the remainder is 1, so 2 is not a divisor of 4111)
4111 / 3 = 1370.3333333333 (the remainder is 1, so 3 is not a divisor of 4111)
...
4111 / 63 = 65.253968253968 (the remainder is 16, so 63 is not a divisor of 4111)
4111 / 64 = 64.234375 (the remainder is 15, so 64 is not a divisor of 4111)