What are the divisors of 722?

1, 2, 19, 38, 361, 722

There is a total of 6 positive divisors.
The sum of these divisors is 1143.
The arithmetic mean is 190.5.

3 even divisors

2, 38, 722

3 odd divisors

1, 19, 361

How to compute the divisors of 722?

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 722 by each of the numbers from 1 to 722 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)

722 / 1 = 722 (the remainder is 0, so 1 is a divisor of 722)
722 / 2 = 361 (the remainder is 0, so 2 is a divisor of 722)
722 / 3 = 240.66666666667 (the remainder is 2, so 3 is not a divisor of 722)
...
722 / 721 = 1.001386962552 (the remainder is 1, so 721 is not a divisor of 722)
722 / 722 = 1 (the remainder is 0, so 722 is a divisor of 722)

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 722 (i.e. 26.870057685089). 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 $\frac{N}{d} = D$ )

722 / 1 = 722 (the remainder is 0, so 1 and 722 are divisors of 722)
722 / 2 = 361 (the remainder is 0, so 2 and 361 are divisors of 722)
722 / 3 = 240.66666666667 (the remainder is 2, so 3 is not a divisor of 722)
...
722 / 25 = 28.88 (the remainder is 22, so 25 is not a divisor of 722)
722 / 26 = 27.769230769231 (the remainder is 20, so 26 is not a divisor of 722)