What are the divisors of 715?

1, 5, 11, 13, 55, 65, 143, 715

There is a total of 8 positive divisors.
The sum of these divisors is 1008.
The arithmetic mean is 126.

8 odd divisors

1, 5, 11, 13, 55, 65, 143, 715

How to compute the divisors of 715?

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 715 by each of the numbers from 1 to 715 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)

715 / 1 = 715 (the remainder is 0, so 1 is a divisor of 715)
715 / 2 = 357.5 (the remainder is 1, so 2 is not a divisor of 715)
715 / 3 = 238.33333333333 (the remainder is 1, so 3 is not a divisor of 715)
...
715 / 714 = 1.0014005602241 (the remainder is 1, so 714 is not a divisor of 715)
715 / 715 = 1 (the remainder is 0, so 715 is a divisor of 715)

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 715 (i.e. 26.739483914242). 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 $\frac{N}{d} = D$ )

715 / 1 = 715 (the remainder is 0, so 1 and 715 are divisors of 715)
715 / 2 = 357.5 (the remainder is 1, so 2 is not a divisor of 715)
715 / 3 = 238.33333333333 (the remainder is 1, so 3 is not a divisor of 715)
...
715 / 25 = 28.6 (the remainder is 15, so 25 is not a divisor of 715)
715 / 26 = 27.5 (the remainder is 13, so 26 is not a divisor of 715)