What are the divisors of 4159?

1, 4159

There is a total of 2 positive divisors.
The sum of these divisors is 4160.
The arithmetic mean is 2080.

2 odd divisors

1, 4159

How to compute the divisors of 4159?

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 4159 by each of the numbers from 1 to 4159 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)

4159 / 1 = 4159 (the remainder is 0, so 1 is a divisor of 4159)
4159 / 2 = 2079.5 (the remainder is 1, so 2 is not a divisor of 4159)
4159 / 3 = 1386.3333333333 (the remainder is 1, so 3 is not a divisor of 4159)
...
4159 / 4158 = 1.0002405002405 (the remainder is 1, so 4158 is not a divisor of 4159)
4159 / 4159 = 1 (the remainder is 0, so 4159 is a divisor of 4159)

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 4159 (i.e. 64.490309349545). 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 $\frac{N}{d} = D$ )

4159 / 1 = 4159 (the remainder is 0, so 1 and 4159 are divisors of 4159)
4159 / 2 = 2079.5 (the remainder is 1, so 2 is not a divisor of 4159)
4159 / 3 = 1386.3333333333 (the remainder is 1, so 3 is not a divisor of 4159)
...
4159 / 63 = 66.015873015873 (the remainder is 1, so 63 is not a divisor of 4159)
4159 / 64 = 64.984375 (the remainder is 63, so 64 is not a divisor of 4159)