What are the divisors of 1519?

1, 7, 31, 49, 217, 1519

There is a total of 6 positive divisors.
The sum of these divisors is 1824.
The arithmetic mean is 304.

6 odd divisors

1, 7, 31, 49, 217, 1519

How to compute the divisors of 1519?

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 1519 by each of the numbers from 1 to 1519 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)

1519 / 1 = 1519 (the remainder is 0, so 1 is a divisor of 1519)
1519 / 2 = 759.5 (the remainder is 1, so 2 is not a divisor of 1519)
1519 / 3 = 506.33333333333 (the remainder is 1, so 3 is not a divisor of 1519)
...
1519 / 1518 = 1.0006587615283 (the remainder is 1, so 1518 is not a divisor of 1519)
1519 / 1519 = 1 (the remainder is 0, so 1519 is a divisor of 1519)

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 1519 (i.e. 38.97435053981). 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 $\frac{N}{d} = D$ )

1519 / 1 = 1519 (the remainder is 0, so 1 and 1519 are divisors of 1519)
1519 / 2 = 759.5 (the remainder is 1, so 2 is not a divisor of 1519)
1519 / 3 = 506.33333333333 (the remainder is 1, so 3 is not a divisor of 1519)
...
1519 / 37 = 41.054054054054 (the remainder is 2, so 37 is not a divisor of 1519)
1519 / 38 = 39.973684210526 (the remainder is 37, so 38 is not a divisor of 1519)