What are the divisors of 745?

1, 5, 149, 745

There is a total of 4 positive divisors.
The sum of these divisors is 900.
The arithmetic mean is 225.

4 odd divisors

1, 5, 149, 745

How to compute the divisors of 745?

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 745 by each of the numbers from 1 to 745 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)

745 / 1 = 745 (the remainder is 0, so 1 is a divisor of 745)
745 / 2 = 372.5 (the remainder is 1, so 2 is not a divisor of 745)
745 / 3 = 248.33333333333 (the remainder is 1, so 3 is not a divisor of 745)
...
745 / 744 = 1.0013440860215 (the remainder is 1, so 744 is not a divisor of 745)
745 / 745 = 1 (the remainder is 0, so 745 is a divisor of 745)

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 745 (i.e. 27.294688127912). 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 $\frac{N}{d} = D$ )

745 / 1 = 745 (the remainder is 0, so 1 and 745 are divisors of 745)
745 / 2 = 372.5 (the remainder is 1, so 2 is not a divisor of 745)
745 / 3 = 248.33333333333 (the remainder is 1, so 3 is not a divisor of 745)
...
745 / 26 = 28.653846153846 (the remainder is 17, so 26 is not a divisor of 745)
745 / 27 = 27.592592592593 (the remainder is 16, so 27 is not a divisor of 745)