What are the divisors of 4205?

1, 5, 29, 145, 841, 4205

There is a total of 6 positive divisors.
The sum of these divisors is 5226.
The arithmetic mean is 871.

6 odd divisors

1, 5, 29, 145, 841, 4205

How to compute the divisors of 4205?

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 4205 by each of the numbers from 1 to 4205 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)

4205 / 1 = 4205 (the remainder is 0, so 1 is a divisor of 4205)
4205 / 2 = 2102.5 (the remainder is 1, so 2 is not a divisor of 4205)
4205 / 3 = 1401.6666666667 (the remainder is 2, so 3 is not a divisor of 4205)
...
4205 / 4204 = 1.0002378686965 (the remainder is 1, so 4204 is not a divisor of 4205)
4205 / 4205 = 1 (the remainder is 0, so 4205 is a divisor of 4205)

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 4205 (i.e. 64.845971347494). 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 $\frac{N}{d} = D$ )

4205 / 1 = 4205 (the remainder is 0, so 1 and 4205 are divisors of 4205)
4205 / 2 = 2102.5 (the remainder is 1, so 2 is not a divisor of 4205)
4205 / 3 = 1401.6666666667 (the remainder is 2, so 3 is not a divisor of 4205)
...
4205 / 63 = 66.746031746032 (the remainder is 47, so 63 is not a divisor of 4205)
4205 / 64 = 65.703125 (the remainder is 45, so 64 is not a divisor of 4205)