What are the divisors of 5225?

1, 5, 11, 19, 25, 55, 95, 209, 275, 475, 1045, 5225

There is a total of 12 positive divisors.
The sum of these divisors is 7440.
The arithmetic mean is 620.

12 odd divisors

1, 5, 11, 19, 25, 55, 95, 209, 275, 475, 1045, 5225

How to compute the divisors of 5225?

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 5225 by each of the numbers from 1 to 5225 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)

5225 / 1 = 5225 (the remainder is 0, so 1 is a divisor of 5225)
5225 / 2 = 2612.5 (the remainder is 1, so 2 is not a divisor of 5225)
5225 / 3 = 1741.6666666667 (the remainder is 2, so 3 is not a divisor of 5225)
...
5225 / 5224 = 1.000191424196 (the remainder is 1, so 5224 is not a divisor of 5225)
5225 / 5225 = 1 (the remainder is 0, so 5225 is a divisor of 5225)

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 5225 (i.e. 72.284161474005). 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 $\frac{N}{d} = D$ )

5225 / 1 = 5225 (the remainder is 0, so 1 and 5225 are divisors of 5225)
5225 / 2 = 2612.5 (the remainder is 1, so 2 is not a divisor of 5225)
5225 / 3 = 1741.6666666667 (the remainder is 2, so 3 is not a divisor of 5225)
...
5225 / 71 = 73.591549295775 (the remainder is 42, so 71 is not a divisor of 5225)
5225 / 72 = 72.569444444444 (the remainder is 41, so 72 is not a divisor of 5225)