What are the divisors of 1749?

1, 3, 11, 33, 53, 159, 583, 1749

There is a total of 8 positive divisors.
The sum of these divisors is 2592.
The arithmetic mean is 324.

8 odd divisors

1, 3, 11, 33, 53, 159, 583, 1749

How to compute the divisors of 1749?

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 1749 by each of the numbers from 1 to 1749 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)

1749 / 1 = 1749 (the remainder is 0, so 1 is a divisor of 1749)
1749 / 2 = 874.5 (the remainder is 1, so 2 is not a divisor of 1749)
1749 / 3 = 583 (the remainder is 0, so 3 is a divisor of 1749)
...
1749 / 1748 = 1.0005720823799 (the remainder is 1, so 1748 is not a divisor of 1749)
1749 / 1749 = 1 (the remainder is 0, so 1749 is a divisor of 1749)

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 1749 (i.e. 41.821047332653). 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 $\frac{N}{d} = D$ )

1749 / 1 = 1749 (the remainder is 0, so 1 and 1749 are divisors of 1749)
1749 / 2 = 874.5 (the remainder is 1, so 2 is not a divisor of 1749)
1749 / 3 = 583 (the remainder is 0, so 3 and 583 are divisors of 1749)
...
1749 / 40 = 43.725 (the remainder is 29, so 40 is not a divisor of 1749)
1749 / 41 = 42.658536585366 (the remainder is 27, so 41 is not a divisor of 1749)