What are the divisors of 1545?

1, 3, 5, 15, 103, 309, 515, 1545

There is a total of 8 positive divisors.
The sum of these divisors is 2496.
The arithmetic mean is 312.

8 odd divisors

1, 3, 5, 15, 103, 309, 515, 1545

How to compute the divisors of 1545?

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 1545 by each of the numbers from 1 to 1545 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)

1545 / 1 = 1545 (the remainder is 0, so 1 is a divisor of 1545)
1545 / 2 = 772.5 (the remainder is 1, so 2 is not a divisor of 1545)
1545 / 3 = 515 (the remainder is 0, so 3 is a divisor of 1545)
...
1545 / 1544 = 1.0006476683938 (the remainder is 1, so 1544 is not a divisor of 1545)
1545 / 1545 = 1 (the remainder is 0, so 1545 is a divisor of 1545)

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 1545 (i.e. 39.306488014067). 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 $\frac{N}{d} = D$ )

1545 / 1 = 1545 (the remainder is 0, so 1 and 1545 are divisors of 1545)
1545 / 2 = 772.5 (the remainder is 1, so 2 is not a divisor of 1545)
1545 / 3 = 515 (the remainder is 0, so 3 and 515 are divisors of 1545)
...
1545 / 38 = 40.657894736842 (the remainder is 25, so 38 is not a divisor of 1545)
1545 / 39 = 39.615384615385 (the remainder is 24, so 39 is not a divisor of 1545)