What are the divisors of 705?

1, 3, 5, 15, 47, 141, 235, 705

There is a total of 8 positive divisors.
The sum of these divisors is 1152.
The arithmetic mean is 144.

8 odd divisors

1, 3, 5, 15, 47, 141, 235, 705

How to compute the divisors of 705?

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 705 by each of the numbers from 1 to 705 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)

705 / 1 = 705 (the remainder is 0, so 1 is a divisor of 705)
705 / 2 = 352.5 (the remainder is 1, so 2 is not a divisor of 705)
705 / 3 = 235 (the remainder is 0, so 3 is a divisor of 705)
...
705 / 704 = 1.0014204545455 (the remainder is 1, so 704 is not a divisor of 705)
705 / 705 = 1 (the remainder is 0, so 705 is a divisor of 705)

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 705 (i.e. 26.551836094704). 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 $\frac{N}{d} = D$ )

705 / 1 = 705 (the remainder is 0, so 1 and 705 are divisors of 705)
705 / 2 = 352.5 (the remainder is 1, so 2 is not a divisor of 705)
705 / 3 = 235 (the remainder is 0, so 3 and 235 are divisors of 705)
...
705 / 25 = 28.2 (the remainder is 5, so 25 is not a divisor of 705)
705 / 26 = 27.115384615385 (the remainder is 3, so 26 is not a divisor of 705)