What are the divisors of 1575?

1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 63, 75, 105, 175, 225, 315, 525, 1575

There is a total of 18 positive divisors.
The sum of these divisors is 3224.
The arithmetic mean is 179.11111111111.

18 odd divisors

1, 3, 5, 7, 9, 15, 21, 25, 35, 45, 63, 75, 105, 175, 225, 315, 525, 1575

How to compute the divisors of 1575?

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 1575 by each of the numbers from 1 to 1575 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)

1575 / 1 = 1575 (the remainder is 0, so 1 is a divisor of 1575)
1575 / 2 = 787.5 (the remainder is 1, so 2 is not a divisor of 1575)
1575 / 3 = 525 (the remainder is 0, so 3 is a divisor of 1575)
...
1575 / 1574 = 1.0006353240152 (the remainder is 1, so 1574 is not a divisor of 1575)
1575 / 1575 = 1 (the remainder is 0, so 1575 is a divisor of 1575)

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 1575 (i.e. 39.686269665969). 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 $\frac{N}{d} = D$ )

1575 / 1 = 1575 (the remainder is 0, so 1 and 1575 are divisors of 1575)
1575 / 2 = 787.5 (the remainder is 1, so 2 is not a divisor of 1575)
1575 / 3 = 525 (the remainder is 0, so 3 and 525 are divisors of 1575)
...
1575 / 38 = 41.447368421053 (the remainder is 17, so 38 is not a divisor of 1575)
1575 / 39 = 40.384615384615 (the remainder is 15, so 39 is not a divisor of 1575)