What are the divisors of 245?

1, 5, 7, 35, 49, 245

There is a total of 6 positive divisors.
The sum of these divisors is 342.
The arithmetic mean is 57.

6 odd divisors

1, 5, 7, 35, 49, 245

How to compute the divisors of 245?

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 245 by each of the numbers from 1 to 245 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)

245 / 1 = 245 (the remainder is 0, so 1 is a divisor of 245)
245 / 2 = 122.5 (the remainder is 1, so 2 is not a divisor of 245)
245 / 3 = 81.666666666667 (the remainder is 2, so 3 is not a divisor of 245)
...
245 / 244 = 1.0040983606557 (the remainder is 1, so 244 is not a divisor of 245)
245 / 245 = 1 (the remainder is 0, so 245 is a divisor of 245)

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 245 (i.e. 15.652475842499). 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 $\frac{N}{d} = D$ )

245 / 1 = 245 (the remainder is 0, so 1 and 245 are divisors of 245)
245 / 2 = 122.5 (the remainder is 1, so 2 is not a divisor of 245)
245 / 3 = 81.666666666667 (the remainder is 2, so 3 is not a divisor of 245)
...
245 / 14 = 17.5 (the remainder is 7, so 14 is not a divisor of 245)
245 / 15 = 16.333333333333 (the remainder is 5, so 15 is not a divisor of 245)