What are the divisors of 4199?

1, 13, 17, 19, 221, 247, 323, 4199

There is a total of 8 positive divisors.
The sum of these divisors is 5040.
The arithmetic mean is 630.

8 odd divisors

1, 13, 17, 19, 221, 247, 323, 4199

How to compute the divisors of 4199?

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 4199 by each of the numbers from 1 to 4199 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)

4199 / 1 = 4199 (the remainder is 0, so 1 is a divisor of 4199)
4199 / 2 = 2099.5 (the remainder is 1, so 2 is not a divisor of 4199)
4199 / 3 = 1399.6666666667 (the remainder is 2, so 3 is not a divisor of 4199)
...
4199 / 4198 = 1.0002382086708 (the remainder is 1, so 4198 is not a divisor of 4199)
4199 / 4199 = 1 (the remainder is 0, so 4199 is a divisor of 4199)

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 4199 (i.e. 64.79969135729). 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 $\frac{N}{d} = D$ )

4199 / 1 = 4199 (the remainder is 0, so 1 and 4199 are divisors of 4199)
4199 / 2 = 2099.5 (the remainder is 1, so 2 is not a divisor of 4199)
4199 / 3 = 1399.6666666667 (the remainder is 2, so 3 is not a divisor of 4199)
...
4199 / 63 = 66.650793650794 (the remainder is 41, so 63 is not a divisor of 4199)
4199 / 64 = 65.609375 (the remainder is 39, so 64 is not a divisor of 4199)