What are the divisors of 4100?

1, 2, 4, 5, 10, 20, 25, 41, 50, 82, 100, 164, 205, 410, 820, 1025, 2050, 4100

There is a total of 18 positive divisors.
The sum of these divisors is 9114.
The arithmetic mean is 506.33333333333.

12 even divisors

2, 4, 10, 20, 50, 82, 100, 164, 410, 820, 2050, 4100

6 odd divisors

1, 5, 25, 41, 205, 1025

How to compute the divisors of 4100?

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 4100 by each of the numbers from 1 to 4100 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)

4100 / 1 = 4100 (the remainder is 0, so 1 is a divisor of 4100)
4100 / 2 = 2050 (the remainder is 0, so 2 is a divisor of 4100)
4100 / 3 = 1366.6666666667 (the remainder is 2, so 3 is not a divisor of 4100)
...
4100 / 4099 = 1.0002439619419 (the remainder is 1, so 4099 is not a divisor of 4100)
4100 / 4100 = 1 (the remainder is 0, so 4100 is a divisor of 4100)

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 4100 (i.e. 64.031242374328). 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 $\frac{N}{d} = D$ )

4100 / 1 = 4100 (the remainder is 0, so 1 and 4100 are divisors of 4100)
4100 / 2 = 2050 (the remainder is 0, so 2 and 2050 are divisors of 4100)
4100 / 3 = 1366.6666666667 (the remainder is 2, so 3 is not a divisor of 4100)
...
4100 / 63 = 65.079365079365 (the remainder is 5, so 63 is not a divisor of 4100)
4100 / 64 = 64.0625 (the remainder is 4, so 64 is not a divisor of 4100)