What are the divisors of 1293?

1, 3, 431, 1293

There is a total of 4 positive divisors.
The sum of these divisors is 1728.
The arithmetic mean is 432.

4 odd divisors

1, 3, 431, 1293

How to compute the divisors of 1293?

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 1293 by each of the numbers from 1 to 1293 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)

1293 / 1 = 1293 (the remainder is 0, so 1 is a divisor of 1293)
1293 / 2 = 646.5 (the remainder is 1, so 2 is not a divisor of 1293)
1293 / 3 = 431 (the remainder is 0, so 3 is a divisor of 1293)
...
1293 / 1292 = 1.000773993808 (the remainder is 1, so 1292 is not a divisor of 1293)
1293 / 1293 = 1 (the remainder is 0, so 1293 is a divisor of 1293)

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 1293 (i.e. 35.95830919273). 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 $\frac{N}{d} = D$ )

1293 / 1 = 1293 (the remainder is 0, so 1 and 1293 are divisors of 1293)
1293 / 2 = 646.5 (the remainder is 1, so 2 is not a divisor of 1293)
1293 / 3 = 431 (the remainder is 0, so 3 and 431 are divisors of 1293)
...
1293 / 34 = 38.029411764706 (the remainder is 1, so 34 is not a divisor of 1293)
1293 / 35 = 36.942857142857 (the remainder is 33, so 35 is not a divisor of 1293)