What are the divisors of 5625?

1, 3, 5, 9, 15, 25, 45, 75, 125, 225, 375, 625, 1125, 1875, 5625

There is a total of 15 positive divisors.
The sum of these divisors is 10153.
The arithmetic mean is 676.86666666667.

15 odd divisors

1, 3, 5, 9, 15, 25, 45, 75, 125, 225, 375, 625, 1125, 1875, 5625

How to compute the divisors of 5625?

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 5625 by each of the numbers from 1 to 5625 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)

5625 / 1 = 5625 (the remainder is 0, so 1 is a divisor of 5625)
5625 / 2 = 2812.5 (the remainder is 1, so 2 is not a divisor of 5625)
5625 / 3 = 1875 (the remainder is 0, so 3 is a divisor of 5625)
...
5625 / 5624 = 1.0001778093883 (the remainder is 1, so 5624 is not a divisor of 5625)
5625 / 5625 = 1 (the remainder is 0, so 5625 is a divisor of 5625)

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 5625 (i.e. 75). 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 $\frac{N}{d} = D$ )

5625 / 1 = 5625 (the remainder is 0, so 1 and 5625 are divisors of 5625)
5625 / 2 = 2812.5 (the remainder is 1, so 2 is not a divisor of 5625)
5625 / 3 = 1875 (the remainder is 0, so 3 and 1875 are divisors of 5625)
...
5625 / 74 = 76.013513513514 (the remainder is 1, so 74 is not a divisor of 5625)
5625 / 75 = 75 (the remainder is 0, so 75 and 75 are divisors of 5625)