What are the divisors of 730?

1, 2, 5, 10, 73, 146, 365, 730

There is a total of 8 positive divisors.
The sum of these divisors is 1332.
The arithmetic mean is 166.5.

4 even divisors

2, 10, 146, 730

4 odd divisors

1, 5, 73, 365

How to compute the divisors of 730?

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 730 by each of the numbers from 1 to 730 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)

730 / 1 = 730 (the remainder is 0, so 1 is a divisor of 730)
730 / 2 = 365 (the remainder is 0, so 2 is a divisor of 730)
730 / 3 = 243.33333333333 (the remainder is 1, so 3 is not a divisor of 730)
...
730 / 729 = 1.0013717421125 (the remainder is 1, so 729 is not a divisor of 730)
730 / 730 = 1 (the remainder is 0, so 730 is a divisor of 730)

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 730 (i.e. 27.018512172213). 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 $\frac{N}{d} = D$ )

730 / 1 = 730 (the remainder is 0, so 1 and 730 are divisors of 730)
730 / 2 = 365 (the remainder is 0, so 2 and 365 are divisors of 730)
730 / 3 = 243.33333333333 (the remainder is 1, so 3 is not a divisor of 730)
...
730 / 26 = 28.076923076923 (the remainder is 2, so 26 is not a divisor of 730)
730 / 27 = 27.037037037037 (the remainder is 1, so 27 is not a divisor of 730)