What are the divisors of 5175?

1, 3, 5, 9, 15, 23, 25, 45, 69, 75, 115, 207, 225, 345, 575, 1035, 1725, 5175

There is a total of 18 positive divisors.
The sum of these divisors is 9672.
The arithmetic mean is 537.33333333333.

18 odd divisors

1, 3, 5, 9, 15, 23, 25, 45, 69, 75, 115, 207, 225, 345, 575, 1035, 1725, 5175

How to compute the divisors of 5175?

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 5175 by each of the numbers from 1 to 5175 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)

5175 / 1 = 5175 (the remainder is 0, so 1 is a divisor of 5175)
5175 / 2 = 2587.5 (the remainder is 1, so 2 is not a divisor of 5175)
5175 / 3 = 1725 (the remainder is 0, so 3 is a divisor of 5175)
...
5175 / 5174 = 1.0001932740626 (the remainder is 1, so 5174 is not a divisor of 5175)
5175 / 5175 = 1 (the remainder is 0, so 5175 is a divisor of 5175)

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 5175 (i.e. 71.937472849691). 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 $\frac{N}{d} = D$ )

5175 / 1 = 5175 (the remainder is 0, so 1 and 5175 are divisors of 5175)
5175 / 2 = 2587.5 (the remainder is 1, so 2 is not a divisor of 5175)
5175 / 3 = 1725 (the remainder is 0, so 3 and 1725 are divisors of 5175)
...
5175 / 70 = 73.928571428571 (the remainder is 65, so 70 is not a divisor of 5175)
5175 / 71 = 72.887323943662 (the remainder is 63, so 71 is not a divisor of 5175)