What are the divisors of 3225?

1, 3, 5, 15, 25, 43, 75, 129, 215, 645, 1075, 3225

There is a total of 12 positive divisors.
The sum of these divisors is 5456.
The arithmetic mean is 454.66666666667.

12 odd divisors

1, 3, 5, 15, 25, 43, 75, 129, 215, 645, 1075, 3225

How to compute the divisors of 3225?

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 3225 by each of the numbers from 1 to 3225 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)

3225 / 1 = 3225 (the remainder is 0, so 1 is a divisor of 3225)
3225 / 2 = 1612.5 (the remainder is 1, so 2 is not a divisor of 3225)
3225 / 3 = 1075 (the remainder is 0, so 3 is a divisor of 3225)
...
3225 / 3224 = 1.0003101736973 (the remainder is 1, so 3224 is not a divisor of 3225)
3225 / 3225 = 1 (the remainder is 0, so 3225 is a divisor of 3225)

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 3225 (i.e. 56.789083458003). 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 $\frac{N}{d} = D$ )

3225 / 1 = 3225 (the remainder is 0, so 1 and 3225 are divisors of 3225)
3225 / 2 = 1612.5 (the remainder is 1, so 2 is not a divisor of 3225)
3225 / 3 = 1075 (the remainder is 0, so 3 and 1075 are divisors of 3225)
...
3225 / 55 = 58.636363636364 (the remainder is 35, so 55 is not a divisor of 3225)
3225 / 56 = 57.589285714286 (the remainder is 33, so 56 is not a divisor of 3225)