What are the divisors of 4225?

1, 5, 13, 25, 65, 169, 325, 845, 4225

There is a total of 9 positive divisors.
The sum of these divisors is 5673.
The arithmetic mean is 630.33333333333.

9 odd divisors

1, 5, 13, 25, 65, 169, 325, 845, 4225

How to compute the divisors of 4225?

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 4225 by each of the numbers from 1 to 4225 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)

4225 / 1 = 4225 (the remainder is 0, so 1 is a divisor of 4225)
4225 / 2 = 2112.5 (the remainder is 1, so 2 is not a divisor of 4225)
4225 / 3 = 1408.3333333333 (the remainder is 1, so 3 is not a divisor of 4225)
...
4225 / 4224 = 1.0002367424242 (the remainder is 1, so 4224 is not a divisor of 4225)
4225 / 4225 = 1 (the remainder is 0, so 4225 is a divisor of 4225)

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 4225 (i.e. 65). 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 $\frac{N}{d} = D$ )

4225 / 1 = 4225 (the remainder is 0, so 1 and 4225 are divisors of 4225)
4225 / 2 = 2112.5 (the remainder is 1, so 2 is not a divisor of 4225)
4225 / 3 = 1408.3333333333 (the remainder is 1, so 3 is not a divisor of 4225)
...
4225 / 64 = 66.015625 (the remainder is 1, so 64 is not a divisor of 4225)
4225 / 65 = 65 (the remainder is 0, so 65 and 65 are divisors of 4225)