What are the divisors of 4137?

1, 3, 7, 21, 197, 591, 1379, 4137

There is a total of 8 positive divisors.
The sum of these divisors is 6336.
The arithmetic mean is 792.

8 odd divisors

1, 3, 7, 21, 197, 591, 1379, 4137

How to compute the divisors of 4137?

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 4137 by each of the numbers from 1 to 4137 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)

4137 / 1 = 4137 (the remainder is 0, so 1 is a divisor of 4137)
4137 / 2 = 2068.5 (the remainder is 1, so 2 is not a divisor of 4137)
4137 / 3 = 1379 (the remainder is 0, so 3 is a divisor of 4137)
...
4137 / 4136 = 1.0002417794971 (the remainder is 1, so 4136 is not a divisor of 4137)
4137 / 4137 = 1 (the remainder is 0, so 4137 is a divisor of 4137)

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 4137 (i.e. 64.319514923544). 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 $\frac{N}{d} = D$ )

4137 / 1 = 4137 (the remainder is 0, so 1 and 4137 are divisors of 4137)
4137 / 2 = 2068.5 (the remainder is 1, so 2 is not a divisor of 4137)
4137 / 3 = 1379 (the remainder is 0, so 3 and 1379 are divisors of 4137)
...
4137 / 63 = 65.666666666667 (the remainder is 42, so 63 is not a divisor of 4137)
4137 / 64 = 64.640625 (the remainder is 41, so 64 is not a divisor of 4137)