What are the divisors of 1743?

1, 3, 7, 21, 83, 249, 581, 1743

There is a total of 8 positive divisors.
The sum of these divisors is 2688.
The arithmetic mean is 336.

8 odd divisors

1, 3, 7, 21, 83, 249, 581, 1743

How to compute the divisors of 1743?

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 1743 by each of the numbers from 1 to 1743 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)

1743 / 1 = 1743 (the remainder is 0, so 1 is a divisor of 1743)
1743 / 2 = 871.5 (the remainder is 1, so 2 is not a divisor of 1743)
1743 / 3 = 581 (the remainder is 0, so 3 is a divisor of 1743)
...
1743 / 1742 = 1.0005740528129 (the remainder is 1, so 1742 is not a divisor of 1743)
1743 / 1743 = 1 (the remainder is 0, so 1743 is a divisor of 1743)

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 1743 (i.e. 41.749251490296). 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 $\frac{N}{d} = D$ )

1743 / 1 = 1743 (the remainder is 0, so 1 and 1743 are divisors of 1743)
1743 / 2 = 871.5 (the remainder is 1, so 2 is not a divisor of 1743)
1743 / 3 = 581 (the remainder is 0, so 3 and 581 are divisors of 1743)
...
1743 / 40 = 43.575 (the remainder is 23, so 40 is not a divisor of 1743)
1743 / 41 = 42.512195121951 (the remainder is 21, so 41 is not a divisor of 1743)