What are the divisors of 1289?

1, 1289

There is a total of 2 positive divisors.
The sum of these divisors is 1290.
The arithmetic mean is 645.

2 odd divisors

1, 1289

How to compute the divisors of 1289?

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 1289 by each of the numbers from 1 to 1289 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)

1289 / 1 = 1289 (the remainder is 0, so 1 is a divisor of 1289)
1289 / 2 = 644.5 (the remainder is 1, so 2 is not a divisor of 1289)
1289 / 3 = 429.66666666667 (the remainder is 2, so 3 is not a divisor of 1289)
...
1289 / 1288 = 1.0007763975155 (the remainder is 1, so 1288 is not a divisor of 1289)
1289 / 1289 = 1 (the remainder is 0, so 1289 is a divisor of 1289)

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 1289 (i.e. 35.902646142032). 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 $\frac{N}{d} = D$ )

1289 / 1 = 1289 (the remainder is 0, so 1 and 1289 are divisors of 1289)
1289 / 2 = 644.5 (the remainder is 1, so 2 is not a divisor of 1289)
1289 / 3 = 429.66666666667 (the remainder is 2, so 3 is not a divisor of 1289)
...
1289 / 34 = 37.911764705882 (the remainder is 31, so 34 is not a divisor of 1289)
1289 / 35 = 36.828571428571 (the remainder is 29, so 35 is not a divisor of 1289)