What are the divisors of 706?

1, 2, 353, 706

There is a total of 4 positive divisors.
The sum of these divisors is 1062.
The arithmetic mean is 265.5.

2 even divisors

2, 706

2 odd divisors

1, 353

How to compute the divisors of 706?

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 706 by each of the numbers from 1 to 706 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)

706 / 1 = 706 (the remainder is 0, so 1 is a divisor of 706)
706 / 2 = 353 (the remainder is 0, so 2 is a divisor of 706)
706 / 3 = 235.33333333333 (the remainder is 1, so 3 is not a divisor of 706)
...
706 / 705 = 1.0014184397163 (the remainder is 1, so 705 is not a divisor of 706)
706 / 706 = 1 (the remainder is 0, so 706 is a divisor of 706)

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 706 (i.e. 26.570660511173). 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 $\frac{N}{d} = D$ )

706 / 1 = 706 (the remainder is 0, so 1 and 706 are divisors of 706)
706 / 2 = 353 (the remainder is 0, so 2 and 353 are divisors of 706)
706 / 3 = 235.33333333333 (the remainder is 1, so 3 is not a divisor of 706)
...
706 / 25 = 28.24 (the remainder is 6, so 25 is not a divisor of 706)
706 / 26 = 27.153846153846 (the remainder is 4, so 26 is not a divisor of 706)