What are the divisors of 1463?

1, 7, 11, 19, 77, 133, 209, 1463

There is a total of 8 positive divisors.
The sum of these divisors is 1920.
The arithmetic mean is 240.

8 odd divisors

1, 7, 11, 19, 77, 133, 209, 1463

How to compute the divisors of 1463?

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 1463 by each of the numbers from 1 to 1463 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)

1463 / 1 = 1463 (the remainder is 0, so 1 is a divisor of 1463)
1463 / 2 = 731.5 (the remainder is 1, so 2 is not a divisor of 1463)
1463 / 3 = 487.66666666667 (the remainder is 2, so 3 is not a divisor of 1463)
...
1463 / 1462 = 1.000683994528 (the remainder is 1, so 1462 is not a divisor of 1463)
1463 / 1463 = 1 (the remainder is 0, so 1463 is a divisor of 1463)

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 1463 (i.e. 38.249182997811). 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 $\frac{N}{d} = D$ )

1463 / 1 = 1463 (the remainder is 0, so 1 and 1463 are divisors of 1463)
1463 / 2 = 731.5 (the remainder is 1, so 2 is not a divisor of 1463)
1463 / 3 = 487.66666666667 (the remainder is 2, so 3 is not a divisor of 1463)
...
1463 / 37 = 39.540540540541 (the remainder is 20, so 37 is not a divisor of 1463)
1463 / 38 = 38.5 (the remainder is 19, so 38 is not a divisor of 1463)