What are the divisors of 1741?

1, 1741

There is a total of 2 positive divisors.
The sum of these divisors is 1742.
The arithmetic mean is 871.

2 odd divisors

1, 1741

How to compute the divisors of 1741?

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 1741 by each of the numbers from 1 to 1741 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)

1741 / 1 = 1741 (the remainder is 0, so 1 is a divisor of 1741)
1741 / 2 = 870.5 (the remainder is 1, so 2 is not a divisor of 1741)
1741 / 3 = 580.33333333333 (the remainder is 1, so 3 is not a divisor of 1741)
...
1741 / 1740 = 1.0005747126437 (the remainder is 1, so 1740 is not a divisor of 1741)
1741 / 1741 = 1 (the remainder is 0, so 1741 is a divisor of 1741)

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 1741 (i.e. 41.72529209005). 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 $\frac{N}{d} = D$ )

1741 / 1 = 1741 (the remainder is 0, so 1 and 1741 are divisors of 1741)
1741 / 2 = 870.5 (the remainder is 1, so 2 is not a divisor of 1741)
1741 / 3 = 580.33333333333 (the remainder is 1, so 3 is not a divisor of 1741)
...
1741 / 40 = 43.525 (the remainder is 21, so 40 is not a divisor of 1741)
1741 / 41 = 42.463414634146 (the remainder is 19, so 41 is not a divisor of 1741)