What are the divisors of 1275?

1, 3, 5, 15, 17, 25, 51, 75, 85, 255, 425, 1275

There is a total of 12 positive divisors.
The sum of these divisors is 2232.
The arithmetic mean is 186.

12 odd divisors

1, 3, 5, 15, 17, 25, 51, 75, 85, 255, 425, 1275

How to compute the divisors of 1275?

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 1275 by each of the numbers from 1 to 1275 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)

1275 / 1 = 1275 (the remainder is 0, so 1 is a divisor of 1275)
1275 / 2 = 637.5 (the remainder is 1, so 2 is not a divisor of 1275)
1275 / 3 = 425 (the remainder is 0, so 3 is a divisor of 1275)
...
1275 / 1274 = 1.0007849293564 (the remainder is 1, so 1274 is not a divisor of 1275)
1275 / 1275 = 1 (the remainder is 0, so 1275 is a divisor of 1275)

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 1275 (i.e. 35.707142142714). 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 $\frac{N}{d} = D$ )

1275 / 1 = 1275 (the remainder is 0, so 1 and 1275 are divisors of 1275)
1275 / 2 = 637.5 (the remainder is 1, so 2 is not a divisor of 1275)
1275 / 3 = 425 (the remainder is 0, so 3 and 425 are divisors of 1275)
...
1275 / 34 = 37.5 (the remainder is 17, so 34 is not a divisor of 1275)
1275 / 35 = 36.428571428571 (the remainder is 15, so 35 is not a divisor of 1275)