What are the divisors of 1721?

1, 1721

There is a total of 2 positive divisors.
The sum of these divisors is 1722.
The arithmetic mean is 861.

2 odd divisors

1, 1721

How to compute the divisors of 1721?

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 1721 by each of the numbers from 1 to 1721 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)

1721 / 1 = 1721 (the remainder is 0, so 1 is a divisor of 1721)
1721 / 2 = 860.5 (the remainder is 1, so 2 is not a divisor of 1721)
1721 / 3 = 573.66666666667 (the remainder is 2, so 3 is not a divisor of 1721)
...
1721 / 1720 = 1.0005813953488 (the remainder is 1, so 1720 is not a divisor of 1721)
1721 / 1721 = 1 (the remainder is 0, so 1721 is a divisor of 1721)

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 1721 (i.e. 41.484937025383). 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 $\frac{N}{d} = D$ )

1721 / 1 = 1721 (the remainder is 0, so 1 and 1721 are divisors of 1721)
1721 / 2 = 860.5 (the remainder is 1, so 2 is not a divisor of 1721)
1721 / 3 = 573.66666666667 (the remainder is 2, so 3 is not a divisor of 1721)
...
1721 / 40 = 43.025 (the remainder is 1, so 40 is not a divisor of 1721)
1721 / 41 = 41.975609756098 (the remainder is 40, so 41 is not a divisor of 1721)