What are the divisors of 525?

1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, 525

There is a total of 12 positive divisors.
The sum of these divisors is 992.
The arithmetic mean is 82.666666666667.

12 odd divisors

1, 3, 5, 7, 15, 21, 25, 35, 75, 105, 175, 525

How to compute the divisors of 525?

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 525 by each of the numbers from 1 to 525 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)

525 / 1 = 525 (the remainder is 0, so 1 is a divisor of 525)
525 / 2 = 262.5 (the remainder is 1, so 2 is not a divisor of 525)
525 / 3 = 175 (the remainder is 0, so 3 is a divisor of 525)
...
525 / 524 = 1.0019083969466 (the remainder is 1, so 524 is not a divisor of 525)
525 / 525 = 1 (the remainder is 0, so 525 is a divisor of 525)

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 525 (i.e. 22.912878474779). 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 $\frac{N}{d} = D$ )

525 / 1 = 525 (the remainder is 0, so 1 and 525 are divisors of 525)
525 / 2 = 262.5 (the remainder is 1, so 2 is not a divisor of 525)
525 / 3 = 175 (the remainder is 0, so 3 and 175 are divisors of 525)
...
525 / 21 = 25 (the remainder is 0, so 21 and 25 are divisors of 525)
525 / 22 = 23.863636363636 (the remainder is 19, so 22 is not a divisor of 525)