What are the divisors of 748?

1, 2, 4, 11, 17, 22, 34, 44, 68, 187, 374, 748

There is a total of 12 positive divisors.
The sum of these divisors is 1512.
The arithmetic mean is 126.

8 even divisors

2, 4, 22, 34, 44, 68, 374, 748

4 odd divisors

1, 11, 17, 187

How to compute the divisors of 748?

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 748 by each of the numbers from 1 to 748 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)

748 / 1 = 748 (the remainder is 0, so 1 is a divisor of 748)
748 / 2 = 374 (the remainder is 0, so 2 is a divisor of 748)
748 / 3 = 249.33333333333 (the remainder is 1, so 3 is not a divisor of 748)
...
748 / 747 = 1.0013386880857 (the remainder is 1, so 747 is not a divisor of 748)
748 / 748 = 1 (the remainder is 0, so 748 is a divisor of 748)

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 748 (i.e. 27.349588662355). 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 $\frac{N}{d} = D$ )

748 / 1 = 748 (the remainder is 0, so 1 and 748 are divisors of 748)
748 / 2 = 374 (the remainder is 0, so 2 and 374 are divisors of 748)
748 / 3 = 249.33333333333 (the remainder is 1, so 3 is not a divisor of 748)
...
748 / 26 = 28.769230769231 (the remainder is 20, so 26 is not a divisor of 748)
748 / 27 = 27.703703703704 (the remainder is 19, so 27 is not a divisor of 748)