What are the divisors of 1729?

1, 7, 13, 19, 91, 133, 247, 1729

There is a total of 8 positive divisors.
The sum of these divisors is 2240.
The arithmetic mean is 280.

8 odd divisors

1, 7, 13, 19, 91, 133, 247, 1729

How to compute the divisors of 1729?

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 1729 by each of the numbers from 1 to 1729 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)

1729 / 1 = 1729 (the remainder is 0, so 1 is a divisor of 1729)
1729 / 2 = 864.5 (the remainder is 1, so 2 is not a divisor of 1729)
1729 / 3 = 576.33333333333 (the remainder is 1, so 3 is not a divisor of 1729)
...
1729 / 1728 = 1.0005787037037 (the remainder is 1, so 1728 is not a divisor of 1729)
1729 / 1729 = 1 (the remainder is 0, so 1729 is a divisor of 1729)

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 1729 (i.e. 41.581245772584). 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 $\frac{N}{d} = D$ )

1729 / 1 = 1729 (the remainder is 0, so 1 and 1729 are divisors of 1729)
1729 / 2 = 864.5 (the remainder is 1, so 2 is not a divisor of 1729)
1729 / 3 = 576.33333333333 (the remainder is 1, so 3 is not a divisor of 1729)
...
1729 / 40 = 43.225 (the remainder is 9, so 40 is not a divisor of 1729)
1729 / 41 = 42.170731707317 (the remainder is 7, so 41 is not a divisor of 1729)