What are the divisors of 1745?

1, 5, 349, 1745

There is a total of 4 positive divisors.
The sum of these divisors is 2100.
The arithmetic mean is 525.

4 odd divisors

1, 5, 349, 1745

How to compute the divisors of 1745?

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 1745 by each of the numbers from 1 to 1745 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)

1745 / 1 = 1745 (the remainder is 0, so 1 is a divisor of 1745)
1745 / 2 = 872.5 (the remainder is 1, so 2 is not a divisor of 1745)
1745 / 3 = 581.66666666667 (the remainder is 2, so 3 is not a divisor of 1745)
...
1745 / 1744 = 1.0005733944954 (the remainder is 1, so 1744 is not a divisor of 1745)
1745 / 1745 = 1 (the remainder is 0, so 1745 is a divisor of 1745)

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 1745 (i.e. 41.773197148411). 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 $\frac{N}{d} = D$ )

1745 / 1 = 1745 (the remainder is 0, so 1 and 1745 are divisors of 1745)
1745 / 2 = 872.5 (the remainder is 1, so 2 is not a divisor of 1745)
1745 / 3 = 581.66666666667 (the remainder is 2, so 3 is not a divisor of 1745)
...
1745 / 40 = 43.625 (the remainder is 25, so 40 is not a divisor of 1745)
1745 / 41 = 42.560975609756 (the remainder is 23, so 41 is not a divisor of 1745)