What are the divisors of 747?

1, 3, 9, 83, 249, 747

There is a total of 6 positive divisors.
The sum of these divisors is 1092.
The arithmetic mean is 182.

6 odd divisors

1, 3, 9, 83, 249, 747

How to compute the divisors of 747?

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 747 by each of the numbers from 1 to 747 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)

747 / 1 = 747 (the remainder is 0, so 1 is a divisor of 747)
747 / 2 = 373.5 (the remainder is 1, so 2 is not a divisor of 747)
747 / 3 = 249 (the remainder is 0, so 3 is a divisor of 747)
...
747 / 746 = 1.0013404825737 (the remainder is 1, so 746 is not a divisor of 747)
747 / 747 = 1 (the remainder is 0, so 747 is a divisor of 747)

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 747 (i.e. 27.331300737433). 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 $\frac{N}{d} = D$ )

747 / 1 = 747 (the remainder is 0, so 1 and 747 are divisors of 747)
747 / 2 = 373.5 (the remainder is 1, so 2 is not a divisor of 747)
747 / 3 = 249 (the remainder is 0, so 3 and 249 are divisors of 747)
...
747 / 26 = 28.730769230769 (the remainder is 19, so 26 is not a divisor of 747)
747 / 27 = 27.666666666667 (the remainder is 18, so 27 is not a divisor of 747)