What are the divisors of 1245?

1, 3, 5, 15, 83, 249, 415, 1245

There is a total of 8 positive divisors.
The sum of these divisors is 2016.
The arithmetic mean is 252.

8 odd divisors

1, 3, 5, 15, 83, 249, 415, 1245

How to compute the divisors of 1245?

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 1245 by each of the numbers from 1 to 1245 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)

1245 / 1 = 1245 (the remainder is 0, so 1 is a divisor of 1245)
1245 / 2 = 622.5 (the remainder is 1, so 2 is not a divisor of 1245)
1245 / 3 = 415 (the remainder is 0, so 3 is a divisor of 1245)
...
1245 / 1244 = 1.0008038585209 (the remainder is 1, so 1244 is not a divisor of 1245)
1245 / 1245 = 1 (the remainder is 0, so 1245 is a divisor of 1245)

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 1245 (i.e. 35.284557528755). 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 $\frac{N}{d} = D$ )

1245 / 1 = 1245 (the remainder is 0, so 1 and 1245 are divisors of 1245)
1245 / 2 = 622.5 (the remainder is 1, so 2 is not a divisor of 1245)
1245 / 3 = 415 (the remainder is 0, so 3 and 415 are divisors of 1245)
...
1245 / 34 = 36.617647058824 (the remainder is 21, so 34 is not a divisor of 1245)
1245 / 35 = 35.571428571429 (the remainder is 20, so 35 is not a divisor of 1245)