What are the divisors of 1746?

1, 2, 3, 6, 9, 18, 97, 194, 291, 582, 873, 1746

There is a total of 12 positive divisors.
The sum of these divisors is 3822.
The arithmetic mean is 318.5.

6 even divisors

2, 6, 18, 194, 582, 1746

6 odd divisors

1, 3, 9, 97, 291, 873

How to compute the divisors of 1746?

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 1746 by each of the numbers from 1 to 1746 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)

1746 / 1 = 1746 (the remainder is 0, so 1 is a divisor of 1746)
1746 / 2 = 873 (the remainder is 0, so 2 is a divisor of 1746)
1746 / 3 = 582 (the remainder is 0, so 3 is a divisor of 1746)
...
1746 / 1745 = 1.0005730659026 (the remainder is 1, so 1745 is not a divisor of 1746)
1746 / 1746 = 1 (the remainder is 0, so 1746 is a divisor of 1746)

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 1746 (i.e. 41.785164831552). 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 $\frac{N}{d} = D$ )

1746 / 1 = 1746 (the remainder is 0, so 1 and 1746 are divisors of 1746)
1746 / 2 = 873 (the remainder is 0, so 2 and 873 are divisors of 1746)
1746 / 3 = 582 (the remainder is 0, so 3 and 582 are divisors of 1746)
...
1746 / 40 = 43.65 (the remainder is 26, so 40 is not a divisor of 1746)
1746 / 41 = 42.585365853659 (the remainder is 24, so 41 is not a divisor of 1746)