What are the divisors of 4167?

1, 3, 9, 463, 1389, 4167

There is a total of 6 positive divisors.
The sum of these divisors is 6032.
The arithmetic mean is 1005.3333333333.

6 odd divisors

1, 3, 9, 463, 1389, 4167

How to compute the divisors of 4167?

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 4167 by each of the numbers from 1 to 4167 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)

4167 / 1 = 4167 (the remainder is 0, so 1 is a divisor of 4167)
4167 / 2 = 2083.5 (the remainder is 1, so 2 is not a divisor of 4167)
4167 / 3 = 1389 (the remainder is 0, so 3 is a divisor of 4167)
...
4167 / 4166 = 1.0002400384061 (the remainder is 1, so 4166 is not a divisor of 4167)
4167 / 4167 = 1 (the remainder is 0, so 4167 is a divisor of 4167)

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 4167 (i.e. 64.55230437405). 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 $\frac{N}{d} = D$ )

4167 / 1 = 4167 (the remainder is 0, so 1 and 4167 are divisors of 4167)
4167 / 2 = 2083.5 (the remainder is 1, so 2 is not a divisor of 4167)
4167 / 3 = 1389 (the remainder is 0, so 3 and 1389 are divisors of 4167)
...
4167 / 63 = 66.142857142857 (the remainder is 9, so 63 is not a divisor of 4167)
4167 / 64 = 65.109375 (the remainder is 7, so 64 is not a divisor of 4167)