What are the divisors of 785?

1, 5, 157, 785

There is a total of 4 positive divisors.
The sum of these divisors is 948.
The arithmetic mean is 237.

4 odd divisors

1, 5, 157, 785

How to compute the divisors of 785?

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 785 by each of the numbers from 1 to 785 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)

785 / 1 = 785 (the remainder is 0, so 1 is a divisor of 785)
785 / 2 = 392.5 (the remainder is 1, so 2 is not a divisor of 785)
785 / 3 = 261.66666666667 (the remainder is 2, so 3 is not a divisor of 785)
...
785 / 784 = 1.0012755102041 (the remainder is 1, so 784 is not a divisor of 785)
785 / 785 = 1 (the remainder is 0, so 785 is a divisor of 785)

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 785 (i.e. 28.017851452244). 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 $\frac{N}{d} = D$ )

785 / 1 = 785 (the remainder is 0, so 1 and 785 are divisors of 785)
785 / 2 = 392.5 (the remainder is 1, so 2 is not a divisor of 785)
785 / 3 = 261.66666666667 (the remainder is 2, so 3 is not a divisor of 785)
...
785 / 27 = 29.074074074074 (the remainder is 2, so 27 is not a divisor of 785)
785 / 28 = 28.035714285714 (the remainder is 1, so 28 is not a divisor of 785)