What are the divisors of 1499?

1, 1499

There is a total of 2 positive divisors.
The sum of these divisors is 1500.
The arithmetic mean is 750.

2 odd divisors

1, 1499

How to compute the divisors of 1499?

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 1499 by each of the numbers from 1 to 1499 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)

1499 / 1 = 1499 (the remainder is 0, so 1 is a divisor of 1499)
1499 / 2 = 749.5 (the remainder is 1, so 2 is not a divisor of 1499)
1499 / 3 = 499.66666666667 (the remainder is 2, so 3 is not a divisor of 1499)
...
1499 / 1498 = 1.0006675567423 (the remainder is 1, so 1498 is not a divisor of 1499)
1499 / 1499 = 1 (the remainder is 0, so 1499 is a divisor of 1499)

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 1499 (i.e. 38.716921365212). 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 $\frac{N}{d} = D$ )

1499 / 1 = 1499 (the remainder is 0, so 1 and 1499 are divisors of 1499)
1499 / 2 = 749.5 (the remainder is 1, so 2 is not a divisor of 1499)
1499 / 3 = 499.66666666667 (the remainder is 2, so 3 is not a divisor of 1499)
...
1499 / 37 = 40.513513513514 (the remainder is 19, so 37 is not a divisor of 1499)
1499 / 38 = 39.447368421053 (the remainder is 17, so 38 is not a divisor of 1499)