What are the divisors of 733?

1, 733

There is a total of 2 positive divisors.
The sum of these divisors is 734.
The arithmetic mean is 367.

2 odd divisors

1, 733

How to compute the divisors of 733?

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 733 by each of the numbers from 1 to 733 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)

733 / 1 = 733 (the remainder is 0, so 1 is a divisor of 733)
733 / 2 = 366.5 (the remainder is 1, so 2 is not a divisor of 733)
733 / 3 = 244.33333333333 (the remainder is 1, so 3 is not a divisor of 733)
...
733 / 732 = 1.0013661202186 (the remainder is 1, so 732 is not a divisor of 733)
733 / 733 = 1 (the remainder is 0, so 733 is a divisor of 733)

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 733 (i.e. 27.073972741362). 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 $\frac{N}{d} = D$ )

733 / 1 = 733 (the remainder is 0, so 1 and 733 are divisors of 733)
733 / 2 = 366.5 (the remainder is 1, so 2 is not a divisor of 733)
733 / 3 = 244.33333333333 (the remainder is 1, so 3 is not a divisor of 733)
...
733 / 26 = 28.192307692308 (the remainder is 5, so 26 is not a divisor of 733)
733 / 27 = 27.148148148148 (the remainder is 4, so 27 is not a divisor of 733)