What are the divisors of 782?

1, 2, 17, 23, 34, 46, 391, 782

There is a total of 8 positive divisors.
The sum of these divisors is 1296.
The arithmetic mean is 162.

4 even divisors

2, 34, 46, 782

4 odd divisors

1, 17, 23, 391

How to compute the divisors of 782?

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 782 by each of the numbers from 1 to 782 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)

782 / 1 = 782 (the remainder is 0, so 1 is a divisor of 782)
782 / 2 = 391 (the remainder is 0, so 2 is a divisor of 782)
782 / 3 = 260.66666666667 (the remainder is 2, so 3 is not a divisor of 782)
...
782 / 781 = 1.0012804097311 (the remainder is 1, so 781 is not a divisor of 782)
782 / 782 = 1 (the remainder is 0, so 782 is a divisor of 782)

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 782 (i.e. 27.964262908219). 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 $\frac{N}{d} = D$ )

782 / 1 = 782 (the remainder is 0, so 1 and 782 are divisors of 782)
782 / 2 = 391 (the remainder is 0, so 2 and 391 are divisors of 782)
782 / 3 = 260.66666666667 (the remainder is 2, so 3 is not a divisor of 782)
...
782 / 26 = 30.076923076923 (the remainder is 2, so 26 is not a divisor of 782)
782 / 27 = 28.962962962963 (the remainder is 26, so 27 is not a divisor of 782)