What are the divisors of 1515?

1, 3, 5, 15, 101, 303, 505, 1515

There is a total of 8 positive divisors.
The sum of these divisors is 2448.
The arithmetic mean is 306.

8 odd divisors

1, 3, 5, 15, 101, 303, 505, 1515

How to compute the divisors of 1515?

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 1515 by each of the numbers from 1 to 1515 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)

1515 / 1 = 1515 (the remainder is 0, so 1 is a divisor of 1515)
1515 / 2 = 757.5 (the remainder is 1, so 2 is not a divisor of 1515)
1515 / 3 = 505 (the remainder is 0, so 3 is a divisor of 1515)
...
1515 / 1514 = 1.0006605019815 (the remainder is 1, so 1514 is not a divisor of 1515)
1515 / 1515 = 1 (the remainder is 0, so 1515 is a divisor of 1515)

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 1515 (i.e. 38.923000912057). 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 $\frac{N}{d} = D$ )

1515 / 1 = 1515 (the remainder is 0, so 1 and 1515 are divisors of 1515)
1515 / 2 = 757.5 (the remainder is 1, so 2 is not a divisor of 1515)
1515 / 3 = 505 (the remainder is 0, so 3 and 505 are divisors of 1515)
...
1515 / 37 = 40.945945945946 (the remainder is 35, so 37 is not a divisor of 1515)
1515 / 38 = 39.868421052632 (the remainder is 33, so 38 is not a divisor of 1515)