What are the divisors of 1715?

1, 5, 7, 35, 49, 245, 343, 1715

There is a total of 8 positive divisors.
The sum of these divisors is 2400.
The arithmetic mean is 300.

8 odd divisors

1, 5, 7, 35, 49, 245, 343, 1715

How to compute the divisors of 1715?

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 1715 by each of the numbers from 1 to 1715 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)

1715 / 1 = 1715 (the remainder is 0, so 1 is a divisor of 1715)
1715 / 2 = 857.5 (the remainder is 1, so 2 is not a divisor of 1715)
1715 / 3 = 571.66666666667 (the remainder is 2, so 3 is not a divisor of 1715)
...
1715 / 1714 = 1.0005834305718 (the remainder is 1, so 1714 is not a divisor of 1715)
1715 / 1715 = 1 (the remainder is 0, so 1715 is a divisor of 1715)

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 1715 (i.e. 41.412558481697). 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 $\frac{N}{d} = D$ )

1715 / 1 = 1715 (the remainder is 0, so 1 and 1715 are divisors of 1715)
1715 / 2 = 857.5 (the remainder is 1, so 2 is not a divisor of 1715)
1715 / 3 = 571.66666666667 (the remainder is 2, so 3 is not a divisor of 1715)
...
1715 / 40 = 42.875 (the remainder is 35, so 40 is not a divisor of 1715)
1715 / 41 = 41.829268292683 (the remainder is 34, so 41 is not a divisor of 1715)