What are the divisors of 4125?

1, 3, 5, 11, 15, 25, 33, 55, 75, 125, 165, 275, 375, 825, 1375, 4125

There is a total of 16 positive divisors.
The sum of these divisors is 7488.
The arithmetic mean is 468.

16 odd divisors

1, 3, 5, 11, 15, 25, 33, 55, 75, 125, 165, 275, 375, 825, 1375, 4125

How to compute the divisors of 4125?

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 4125 by each of the numbers from 1 to 4125 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)

4125 / 1 = 4125 (the remainder is 0, so 1 is a divisor of 4125)
4125 / 2 = 2062.5 (the remainder is 1, so 2 is not a divisor of 4125)
4125 / 3 = 1375 (the remainder is 0, so 3 is a divisor of 4125)
...
4125 / 4124 = 1.0002424830262 (the remainder is 1, so 4124 is not a divisor of 4125)
4125 / 4125 = 1 (the remainder is 0, so 4125 is a divisor of 4125)

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 4125 (i.e. 64.226162893326). 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 $\frac{N}{d} = D$ )

4125 / 1 = 4125 (the remainder is 0, so 1 and 4125 are divisors of 4125)
4125 / 2 = 2062.5 (the remainder is 1, so 2 is not a divisor of 4125)
4125 / 3 = 1375 (the remainder is 0, so 3 and 1375 are divisors of 4125)
...
4125 / 63 = 65.47619047619 (the remainder is 30, so 63 is not a divisor of 4125)
4125 / 64 = 64.453125 (the remainder is 29, so 64 is not a divisor of 4125)