What are the divisors of 975?

1, 3, 5, 13, 15, 25, 39, 65, 75, 195, 325, 975

There is a total of 12 positive divisors.
The sum of these divisors is 1736.
The arithmetic mean is 144.66666666667.

12 odd divisors

1, 3, 5, 13, 15, 25, 39, 65, 75, 195, 325, 975

How to compute the divisors of 975?

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 975 by each of the numbers from 1 to 975 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)

975 / 1 = 975 (the remainder is 0, so 1 is a divisor of 975)
975 / 2 = 487.5 (the remainder is 1, so 2 is not a divisor of 975)
975 / 3 = 325 (the remainder is 0, so 3 is a divisor of 975)
...
975 / 974 = 1.0010266940452 (the remainder is 1, so 974 is not a divisor of 975)
975 / 975 = 1 (the remainder is 0, so 975 is a divisor of 975)

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 975 (i.e. 31.224989991992). 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 $\frac{N}{d} = D$ )

975 / 1 = 975 (the remainder is 0, so 1 and 975 are divisors of 975)
975 / 2 = 487.5 (the remainder is 1, so 2 is not a divisor of 975)
975 / 3 = 325 (the remainder is 0, so 3 and 325 are divisors of 975)
...
975 / 30 = 32.5 (the remainder is 15, so 30 is not a divisor of 975)
975 / 31 = 31.451612903226 (the remainder is 14, so 31 is not a divisor of 975)