What are the divisors of 4181?

1, 37, 113, 4181

There is a total of 4 positive divisors.
The sum of these divisors is 4332.
The arithmetic mean is 1083.

4 odd divisors

1, 37, 113, 4181

How to compute the divisors of 4181?

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 4181 by each of the numbers from 1 to 4181 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)

4181 / 1 = 4181 (the remainder is 0, so 1 is a divisor of 4181)
4181 / 2 = 2090.5 (the remainder is 1, so 2 is not a divisor of 4181)
4181 / 3 = 1393.6666666667 (the remainder is 2, so 3 is not a divisor of 4181)
...
4181 / 4180 = 1.0002392344498 (the remainder is 1, so 4180 is not a divisor of 4181)
4181 / 4181 = 1 (the remainder is 0, so 4181 is a divisor of 4181)

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 4181 (i.e. 64.660652641309). 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 $\frac{N}{d} = D$ )

4181 / 1 = 4181 (the remainder is 0, so 1 and 4181 are divisors of 4181)
4181 / 2 = 2090.5 (the remainder is 1, so 2 is not a divisor of 4181)
4181 / 3 = 1393.6666666667 (the remainder is 2, so 3 is not a divisor of 4181)
...
4181 / 63 = 66.365079365079 (the remainder is 23, so 63 is not a divisor of 4181)
4181 / 64 = 65.328125 (the remainder is 21, so 64 is not a divisor of 4181)