What are the divisors of 754?

1, 2, 13, 26, 29, 58, 377, 754

There is a total of 8 positive divisors.
The sum of these divisors is 1260.
The arithmetic mean is 157.5.

4 even divisors

2, 26, 58, 754

4 odd divisors

1, 13, 29, 377

How to compute the divisors of 754?

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 754 by each of the numbers from 1 to 754 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)

754 / 1 = 754 (the remainder is 0, so 1 is a divisor of 754)
754 / 2 = 377 (the remainder is 0, so 2 is a divisor of 754)
754 / 3 = 251.33333333333 (the remainder is 1, so 3 is not a divisor of 754)
...
754 / 753 = 1.0013280212483 (the remainder is 1, so 753 is not a divisor of 754)
754 / 754 = 1 (the remainder is 0, so 754 is a divisor of 754)

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 754 (i.e. 27.459060435492). 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 $\frac{N}{d} = D$ )

754 / 1 = 754 (the remainder is 0, so 1 and 754 are divisors of 754)
754 / 2 = 377 (the remainder is 0, so 2 and 377 are divisors of 754)
754 / 3 = 251.33333333333 (the remainder is 1, so 3 is not a divisor of 754)
...
754 / 26 = 29 (the remainder is 0, so 26 and 29 are divisors of 754)
754 / 27 = 27.925925925926 (the remainder is 25, so 27 is not a divisor of 754)