What are the divisors of 750?

1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 125, 150, 250, 375, 750

There is a total of 16 positive divisors.
The sum of these divisors is 1872.
The arithmetic mean is 117.

8 even divisors

2, 6, 10, 30, 50, 150, 250, 750

8 odd divisors

1, 3, 5, 15, 25, 75, 125, 375

How to compute the divisors of 750?

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 750 by each of the numbers from 1 to 750 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)

750 / 1 = 750 (the remainder is 0, so 1 is a divisor of 750)
750 / 2 = 375 (the remainder is 0, so 2 is a divisor of 750)
750 / 3 = 250 (the remainder is 0, so 3 is a divisor of 750)
...
750 / 749 = 1.0013351134846 (the remainder is 1, so 749 is not a divisor of 750)
750 / 750 = 1 (the remainder is 0, so 750 is a divisor of 750)

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 750 (i.e. 27.386127875258). 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 $\frac{N}{d} = D$ )

750 / 1 = 750 (the remainder is 0, so 1 and 750 are divisors of 750)
750 / 2 = 375 (the remainder is 0, so 2 and 375 are divisors of 750)
750 / 3 = 250 (the remainder is 0, so 3 and 250 are divisors of 750)
...
750 / 26 = 28.846153846154 (the remainder is 22, so 26 is not a divisor of 750)
750 / 27 = 27.777777777778 (the remainder is 21, so 27 is not a divisor of 750)