What are the divisors of 4194?

1, 2, 3, 6, 9, 18, 233, 466, 699, 1398, 2097, 4194

There is a total of 12 positive divisors.
The sum of these divisors is 9126.
The arithmetic mean is 760.5.

6 even divisors

2, 6, 18, 466, 1398, 4194

6 odd divisors

1, 3, 9, 233, 699, 2097

How to compute the divisors of 4194?

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 4194 by each of the numbers from 1 to 4194 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)

4194 / 1 = 4194 (the remainder is 0, so 1 is a divisor of 4194)
4194 / 2 = 2097 (the remainder is 0, so 2 is a divisor of 4194)
4194 / 3 = 1398 (the remainder is 0, so 3 is a divisor of 4194)
...
4194 / 4193 = 1.000238492726 (the remainder is 1, so 4193 is not a divisor of 4194)
4194 / 4194 = 1 (the remainder is 0, so 4194 is a divisor of 4194)

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 4194 (i.e. 64.761099434769). 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 $\frac{N}{d} = D$ )

4194 / 1 = 4194 (the remainder is 0, so 1 and 4194 are divisors of 4194)
4194 / 2 = 2097 (the remainder is 0, so 2 and 2097 are divisors of 4194)
4194 / 3 = 1398 (the remainder is 0, so 3 and 1398 are divisors of 4194)
...
4194 / 63 = 66.571428571429 (the remainder is 36, so 63 is not a divisor of 4194)
4194 / 64 = 65.53125 (the remainder is 34, so 64 is not a divisor of 4194)