What are the divisors of 4127?

1, 4127

There is a total of 2 positive divisors.
The sum of these divisors is 4128.
The arithmetic mean is 2064.

2 odd divisors

1, 4127

How to compute the divisors of 4127?

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 4127 by each of the numbers from 1 to 4127 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)

4127 / 1 = 4127 (the remainder is 0, so 1 is a divisor of 4127)
4127 / 2 = 2063.5 (the remainder is 1, so 2 is not a divisor of 4127)
4127 / 3 = 1375.6666666667 (the remainder is 2, so 3 is not a divisor of 4127)
...
4127 / 4126 = 1.0002423654872 (the remainder is 1, so 4126 is not a divisor of 4127)
4127 / 4127 = 1 (the remainder is 0, so 4127 is a divisor of 4127)

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 4127 (i.e. 64.241730985396). 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 $\frac{N}{d} = D$ )

4127 / 1 = 4127 (the remainder is 0, so 1 and 4127 are divisors of 4127)
4127 / 2 = 2063.5 (the remainder is 1, so 2 is not a divisor of 4127)
4127 / 3 = 1375.6666666667 (the remainder is 2, so 3 is not a divisor of 4127)
...
4127 / 63 = 65.507936507937 (the remainder is 32, so 63 is not a divisor of 4127)
4127 / 64 = 64.484375 (the remainder is 31, so 64 is not a divisor of 4127)