What are the divisors of 712?

1, 2, 4, 8, 89, 178, 356, 712

There is a total of 8 positive divisors.
The sum of these divisors is 1350.
The arithmetic mean is 168.75.

6 even divisors

2, 4, 8, 178, 356, 712

2 odd divisors

1, 89

How to compute the divisors of 712?

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 712 by each of the numbers from 1 to 712 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)

712 / 1 = 712 (the remainder is 0, so 1 is a divisor of 712)
712 / 2 = 356 (the remainder is 0, so 2 is a divisor of 712)
712 / 3 = 237.33333333333 (the remainder is 1, so 3 is not a divisor of 712)
...
712 / 711 = 1.0014064697609 (the remainder is 1, so 711 is not a divisor of 712)
712 / 712 = 1 (the remainder is 0, so 712 is a divisor of 712)

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 712 (i.e. 26.683328128253). 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 $\frac{N}{d} = D$ )

712 / 1 = 712 (the remainder is 0, so 1 and 712 are divisors of 712)
712 / 2 = 356 (the remainder is 0, so 2 and 356 are divisors of 712)
712 / 3 = 237.33333333333 (the remainder is 1, so 3 is not a divisor of 712)
...
712 / 25 = 28.48 (the remainder is 12, so 25 is not a divisor of 712)
712 / 26 = 27.384615384615 (the remainder is 10, so 26 is not a divisor of 712)