What are the divisors of 3975?

1, 3, 5, 15, 25, 53, 75, 159, 265, 795, 1325, 3975

There is a total of 12 positive divisors.
The sum of these divisors is 6696.
The arithmetic mean is 558.

12 odd divisors

1, 3, 5, 15, 25, 53, 75, 159, 265, 795, 1325, 3975

How to compute the divisors of 3975?

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 3975 by each of the numbers from 1 to 3975 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)

3975 / 1 = 3975 (the remainder is 0, so 1 is a divisor of 3975)
3975 / 2 = 1987.5 (the remainder is 1, so 2 is not a divisor of 3975)
3975 / 3 = 1325 (the remainder is 0, so 3 is a divisor of 3975)
...
3975 / 3974 = 1.0002516356316 (the remainder is 1, so 3974 is not a divisor of 3975)
3975 / 3975 = 1 (the remainder is 0, so 3975 is a divisor of 3975)

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 3975 (i.e. 63.047601064592). 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 $\frac{N}{d} = D$ )

3975 / 1 = 3975 (the remainder is 0, so 1 and 3975 are divisors of 3975)
3975 / 2 = 1987.5 (the remainder is 1, so 2 is not a divisor of 3975)
3975 / 3 = 1325 (the remainder is 0, so 3 and 1325 are divisors of 3975)
...
3975 / 62 = 64.112903225806 (the remainder is 7, so 62 is not a divisor of 3975)
3975 / 63 = 63.095238095238 (the remainder is 6, so 63 is not a divisor of 3975)