What are the divisors of 1035?

1, 3, 5, 9, 15, 23, 45, 69, 115, 207, 345, 1035

There is a total of 12 positive divisors.
The sum of these divisors is 1872.
The arithmetic mean is 156.

12 odd divisors

1, 3, 5, 9, 15, 23, 45, 69, 115, 207, 345, 1035

How to compute the divisors of 1035?

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 1035 by each of the numbers from 1 to 1035 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)

1035 / 1 = 1035 (the remainder is 0, so 1 is a divisor of 1035)
1035 / 2 = 517.5 (the remainder is 1, so 2 is not a divisor of 1035)
1035 / 3 = 345 (the remainder is 0, so 3 is a divisor of 1035)
...
1035 / 1034 = 1.0009671179884 (the remainder is 1, so 1034 is not a divisor of 1035)
1035 / 1035 = 1 (the remainder is 0, so 1035 is a divisor of 1035)

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 1035 (i.e. 32.171415884291). 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 $\frac{N}{d} = D$ )

1035 / 1 = 1035 (the remainder is 0, so 1 and 1035 are divisors of 1035)
1035 / 2 = 517.5 (the remainder is 1, so 2 is not a divisor of 1035)
1035 / 3 = 345 (the remainder is 0, so 3 and 345 are divisors of 1035)
...
1035 / 31 = 33.387096774194 (the remainder is 12, so 31 is not a divisor of 1035)
1035 / 32 = 32.34375 (the remainder is 11, so 32 is not a divisor of 1035)