What are the divisors of 741?

1, 3, 13, 19, 39, 57, 247, 741

There is a total of 8 positive divisors.
The sum of these divisors is 1120.
The arithmetic mean is 140.

8 odd divisors

1, 3, 13, 19, 39, 57, 247, 741

How to compute the divisors of 741?

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 741 by each of the numbers from 1 to 741 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)

741 / 1 = 741 (the remainder is 0, so 1 is a divisor of 741)
741 / 2 = 370.5 (the remainder is 1, so 2 is not a divisor of 741)
741 / 3 = 247 (the remainder is 0, so 3 is a divisor of 741)
...
741 / 740 = 1.0013513513514 (the remainder is 1, so 740 is not a divisor of 741)
741 / 741 = 1 (the remainder is 0, so 741 is a divisor of 741)

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 741 (i.e. 27.221315177632). 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 $\frac{N}{d} = D$ )

741 / 1 = 741 (the remainder is 0, so 1 and 741 are divisors of 741)
741 / 2 = 370.5 (the remainder is 1, so 2 is not a divisor of 741)
741 / 3 = 247 (the remainder is 0, so 3 and 247 are divisors of 741)
...
741 / 26 = 28.5 (the remainder is 13, so 26 is not a divisor of 741)
741 / 27 = 27.444444444444 (the remainder is 12, so 27 is not a divisor of 741)