What are the divisors of 4119?

1, 3, 1373, 4119

There is a total of 4 positive divisors.
The sum of these divisors is 5496.
The arithmetic mean is 1374.

4 odd divisors

1, 3, 1373, 4119

How to compute the divisors of 4119?

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 4119 by each of the numbers from 1 to 4119 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)

4119 / 1 = 4119 (the remainder is 0, so 1 is a divisor of 4119)
4119 / 2 = 2059.5 (the remainder is 1, so 2 is not a divisor of 4119)
4119 / 3 = 1373 (the remainder is 0, so 3 is a divisor of 4119)
...
4119 / 4118 = 1.0002428363283 (the remainder is 1, so 4118 is not a divisor of 4119)
4119 / 4119 = 1 (the remainder is 0, so 4119 is a divisor of 4119)

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 4119 (i.e. 64.17943595888). 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 $\frac{N}{d} = D$ )

4119 / 1 = 4119 (the remainder is 0, so 1 and 4119 are divisors of 4119)
4119 / 2 = 2059.5 (the remainder is 1, so 2 is not a divisor of 4119)
4119 / 3 = 1373 (the remainder is 0, so 3 and 1373 are divisors of 4119)
...
4119 / 63 = 65.380952380952 (the remainder is 24, so 63 is not a divisor of 4119)
4119 / 64 = 64.359375 (the remainder is 23, so 64 is not a divisor of 4119)