What are the divisors of 731?

1, 17, 43, 731

There is a total of 4 positive divisors.
The sum of these divisors is 792.
The arithmetic mean is 198.

4 odd divisors

1, 17, 43, 731

How to compute the divisors of 731?

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 731 by each of the numbers from 1 to 731 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)

731 / 1 = 731 (the remainder is 0, so 1 is a divisor of 731)
731 / 2 = 365.5 (the remainder is 1, so 2 is not a divisor of 731)
731 / 3 = 243.66666666667 (the remainder is 2, so 3 is not a divisor of 731)
...
731 / 730 = 1.0013698630137 (the remainder is 1, so 730 is not a divisor of 731)
731 / 731 = 1 (the remainder is 0, so 731 is a divisor of 731)

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 731 (i.e. 27.037011669192). 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 $\frac{N}{d} = D$ )

731 / 1 = 731 (the remainder is 0, so 1 and 731 are divisors of 731)
731 / 2 = 365.5 (the remainder is 1, so 2 is not a divisor of 731)
731 / 3 = 243.66666666667 (the remainder is 2, so 3 is not a divisor of 731)
...
731 / 26 = 28.115384615385 (the remainder is 3, so 26 is not a divisor of 731)
731 / 27 = 27.074074074074 (the remainder is 2, so 27 is not a divisor of 731)