What are the divisors of 1744?

1, 2, 4, 8, 16, 109, 218, 436, 872, 1744

There is a total of 10 positive divisors.
The sum of these divisors is 3410.
The arithmetic mean is 341.

8 even divisors

2, 4, 8, 16, 218, 436, 872, 1744

2 odd divisors

1, 109

How to compute the divisors of 1744?

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 1744 by each of the numbers from 1 to 1744 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)

1744 / 1 = 1744 (the remainder is 0, so 1 is a divisor of 1744)
1744 / 2 = 872 (the remainder is 0, so 2 is a divisor of 1744)
1744 / 3 = 581.33333333333 (the remainder is 1, so 3 is not a divisor of 1744)
...
1744 / 1743 = 1.0005737234653 (the remainder is 1, so 1743 is not a divisor of 1744)
1744 / 1744 = 1 (the remainder is 0, so 1744 is a divisor of 1744)

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 1744 (i.e. 41.761226035642). 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 $\frac{N}{d} = D$ )

1744 / 1 = 1744 (the remainder is 0, so 1 and 1744 are divisors of 1744)
1744 / 2 = 872 (the remainder is 0, so 2 and 872 are divisors of 1744)
1744 / 3 = 581.33333333333 (the remainder is 1, so 3 is not a divisor of 1744)
...
1744 / 40 = 43.6 (the remainder is 24, so 40 is not a divisor of 1744)
1744 / 41 = 42.536585365854 (the remainder is 22, so 41 is not a divisor of 1744)