What are the divisors of 1725?

1, 3, 5, 15, 23, 25, 69, 75, 115, 345, 575, 1725

There is a total of 12 positive divisors.
The sum of these divisors is 2976.
The arithmetic mean is 248.

12 odd divisors

1, 3, 5, 15, 23, 25, 69, 75, 115, 345, 575, 1725

How to compute the divisors of 1725?

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 1725 by each of the numbers from 1 to 1725 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)

1725 / 1 = 1725 (the remainder is 0, so 1 is a divisor of 1725)
1725 / 2 = 862.5 (the remainder is 1, so 2 is not a divisor of 1725)
1725 / 3 = 575 (the remainder is 0, so 3 is a divisor of 1725)
...
1725 / 1724 = 1.0005800464037 (the remainder is 1, so 1724 is not a divisor of 1725)
1725 / 1725 = 1 (the remainder is 0, so 1725 is a divisor of 1725)

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 1725 (i.e. 41.53311931459). 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 $\frac{N}{d} = D$ )

1725 / 1 = 1725 (the remainder is 0, so 1 and 1725 are divisors of 1725)
1725 / 2 = 862.5 (the remainder is 1, so 2 is not a divisor of 1725)
1725 / 3 = 575 (the remainder is 0, so 3 and 575 are divisors of 1725)
...
1725 / 40 = 43.125 (the remainder is 5, so 40 is not a divisor of 1725)
1725 / 41 = 42.073170731707 (the remainder is 3, so 41 is not a divisor of 1725)