What are the divisors of 1225?

1, 5, 7, 25, 35, 49, 175, 245, 1225

There is a total of 9 positive divisors.
The sum of these divisors is 1767.
The arithmetic mean is 196.33333333333.

9 odd divisors

1, 5, 7, 25, 35, 49, 175, 245, 1225

How to compute the divisors of 1225?

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 1225 by each of the numbers from 1 to 1225 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)

1225 / 1 = 1225 (the remainder is 0, so 1 is a divisor of 1225)
1225 / 2 = 612.5 (the remainder is 1, so 2 is not a divisor of 1225)
1225 / 3 = 408.33333333333 (the remainder is 1, so 3 is not a divisor of 1225)
...
1225 / 1224 = 1.0008169934641 (the remainder is 1, so 1224 is not a divisor of 1225)
1225 / 1225 = 1 (the remainder is 0, so 1225 is a divisor of 1225)

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 1225 (i.e. 35). 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 $\frac{N}{d} = D$ )

1225 / 1 = 1225 (the remainder is 0, so 1 and 1225 are divisors of 1225)
1225 / 2 = 612.5 (the remainder is 1, so 2 is not a divisor of 1225)
1225 / 3 = 408.33333333333 (the remainder is 1, so 3 is not a divisor of 1225)
...
1225 / 34 = 36.029411764706 (the remainder is 1, so 34 is not a divisor of 1225)
1225 / 35 = 35 (the remainder is 0, so 35 and 35 are divisors of 1225)