What are the divisors of 3825?

1, 3, 5, 9, 15, 17, 25, 45, 51, 75, 85, 153, 225, 255, 425, 765, 1275, 3825

There is a total of 18 positive divisors.
The sum of these divisors is 7254.
The arithmetic mean is 403.

18 odd divisors

1, 3, 5, 9, 15, 17, 25, 45, 51, 75, 85, 153, 225, 255, 425, 765, 1275, 3825

How to compute the divisors of 3825?

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 3825 by each of the numbers from 1 to 3825 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)

3825 / 1 = 3825 (the remainder is 0, so 1 is a divisor of 3825)
3825 / 2 = 1912.5 (the remainder is 1, so 2 is not a divisor of 3825)
3825 / 3 = 1275 (the remainder is 0, so 3 is a divisor of 3825)
...
3825 / 3824 = 1.0002615062762 (the remainder is 1, so 3824 is not a divisor of 3825)
3825 / 3825 = 1 (the remainder is 0, so 3825 is a divisor of 3825)

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 3825 (i.e. 61.846584384265). 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 $\frac{N}{d} = D$ )

3825 / 1 = 3825 (the remainder is 0, so 1 and 3825 are divisors of 3825)
3825 / 2 = 1912.5 (the remainder is 1, so 2 is not a divisor of 3825)
3825 / 3 = 1275 (the remainder is 0, so 3 and 1275 are divisors of 3825)
...
3825 / 60 = 63.75 (the remainder is 45, so 60 is not a divisor of 3825)
3825 / 61 = 62.704918032787 (the remainder is 43, so 61 is not a divisor of 3825)