What are the divisors of 749?

1, 7, 107, 749

There is a total of 4 positive divisors.
The sum of these divisors is 864.
The arithmetic mean is 216.

4 odd divisors

1, 7, 107, 749

How to compute the divisors of 749?

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 749 by each of the numbers from 1 to 749 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)

749 / 1 = 749 (the remainder is 0, so 1 is a divisor of 749)
749 / 2 = 374.5 (the remainder is 1, so 2 is not a divisor of 749)
749 / 3 = 249.66666666667 (the remainder is 2, so 3 is not a divisor of 749)
...
749 / 748 = 1.0013368983957 (the remainder is 1, so 748 is not a divisor of 749)
749 / 749 = 1 (the remainder is 0, so 749 is a divisor of 749)

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 749 (i.e. 27.367864366808). 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 $\frac{N}{d} = D$ )

749 / 1 = 749 (the remainder is 0, so 1 and 749 are divisors of 749)
749 / 2 = 374.5 (the remainder is 1, so 2 is not a divisor of 749)
749 / 3 = 249.66666666667 (the remainder is 2, so 3 is not a divisor of 749)
...
749 / 26 = 28.807692307692 (the remainder is 21, so 26 is not a divisor of 749)
749 / 27 = 27.740740740741 (the remainder is 20, so 27 is not a divisor of 749)