What are the divisors of 753?

1, 3, 251, 753

There is a total of 4 positive divisors.
The sum of these divisors is 1008.
The arithmetic mean is 252.

4 odd divisors

1, 3, 251, 753

How to compute the divisors of 753?

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 753 by each of the numbers from 1 to 753 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)

753 / 1 = 753 (the remainder is 0, so 1 is a divisor of 753)
753 / 2 = 376.5 (the remainder is 1, so 2 is not a divisor of 753)
753 / 3 = 251 (the remainder is 0, so 3 is a divisor of 753)
...
753 / 752 = 1.001329787234 (the remainder is 1, so 752 is not a divisor of 753)
753 / 753 = 1 (the remainder is 0, so 753 is a divisor of 753)

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 753 (i.e. 27.440845468024). 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 $\frac{N}{d} = D$ )

753 / 1 = 753 (the remainder is 0, so 1 and 753 are divisors of 753)
753 / 2 = 376.5 (the remainder is 1, so 2 is not a divisor of 753)
753 / 3 = 251 (the remainder is 0, so 3 and 251 are divisors of 753)
...
753 / 26 = 28.961538461538 (the remainder is 25, so 26 is not a divisor of 753)
753 / 27 = 27.888888888889 (the remainder is 24, so 27 is not a divisor of 753)