optimal algorithm for particular divisors - php

Question

Not a duplicate of-
optimal algorithm for finding unique divisors
I came across this problem. I am not able to find an optimal algorithm.
The problem is :
Given a list L of natural numbers(number can be really large) and a number N, what's the optimal algorithm to determine the number of divisors of N which doesn't not divide any of the numbers present in the list L. Numbers in the list can be repetitive ie, one number can occur more than once.
Observation:
Divisors of some divisor d of N are also divisors of N.
MY approach was :
Find the divisors of N.
Sort L in reverse order(largest element being 1st element).
foreach divisor d of N, I check whether it divides any element in the list or not.(stop when you come to check for an element less than d in the list, as the list is sorted)
If d divides some number in the list L, then I don't check for any divisor of d, that is, I skip this checking.
Ultimately, left divisors which were neither divided any number in the list nor skipped are counted. This count is the final answer.
But this algorithm is not optimal for this problem.
Any ideas for a better algorithm?

Answer 1

What you need to look into is : co-primes (or relatively primes)
In number theory, a branch of mathematics, two integers a and b are
said to be coprime (also spelled co-prime) or relatively prime if the
only positive integer that evenly divides both of them is 1.
So to "transcode" your problem :
You basically want to find the Number of coprimes of N from the L list.
When a and b are co-primes?
If two numbers are relatively prime then their greatest common divisor (GCD)
is 1
Example code (for GCD) in PHP :
<?php
$gcd = gmp_gcd("12", "21");
echo gmp_strval($gcd) . "\n";
?>
Simply put :
$count = 0
Foreach element e in list L : calculate the GCD(e,N)
Is their GCD=1? If yes, they are coprime (so N and e have no common divisors). Count it up. $count++
And that's all there is to it.

Answer 2

First, factorize n and represent it in the following way: p1:k1, p2:k2,..., pm:km such that p1,p2,... are all primes and n=p1^k1 * p2^k2 ....
Now, iterate over r1, r2, r3,..., rm such that r1<=k1, r2<=k2, ..., rm<=km and check if p1^r1*p2^r2...*pm^rm divides any number in L. If not increment count by 1.
Optimization: Pick a value for r1. See if p1^r1 divides any number in L. If yes, then pick a number for r2 and so on. If p1^r1 does not divide any number in L, then increment count by (k2+1)(k3+1)..*(km+1).
Example N=72, L=[4, 5, 9, 12, 15, 20]:
Writing N as a primal product: 2:3, 3:2 (2^3*3*2 = 72).
p1=2, p2=3, k1=3, k2=2
count=0
r1=0:
r2=0:
Divides 4
r1=0:
r2=1:
Divides 9
r1=0:
r2=2:
Divides 9
r1=1:
r2=0:
Divides 4
r1=1:
r2=1:
Divides 12
r1=1:
r2=2:
L not divisible by 18. Count+=1 = 1
r1=2:
r2=0:
Divides 4
r1=2:
r2=1:
Divides 12
r1=2:
r2=2:
L not divisible by 36. Count+=1 = 2
r1=3:
r2=0:
L not divisible by 8. Count+=(k2+1) +=(2+1) = 5

Answer 3

<?php
class Divisors {
public $factor = array();
public function __construct($num) {
$this->num = $num;
}
// count number of divisors of a number
public function countDivisors() {
if ($this->num == 1) return 1;
$this->_primefactors();
$array_primes = array_count_values($this->factor);
$divisors = 1;
foreach($array_primes as $power) {
$divisors *= ++$power;
}
return $divisors;
}
// prime factors decomposer
private function _primefactors() {
$this->factor = array();
$run = true;
while($run && #$this->factor[0] != $this->num) {
$run = $this->_getFactors();
}
}
// get all factors of the number
private function _getFactors() {
if($this->num == 1) {
return ;
}
$root = ceil(sqrt($this->num)) + 1;
$i = 2;
while($i <= $root) {
if($this->num % $i == 0) {
$this->factor[] = $i;
$this->num = $this->num / $i;
return true;
}
$i++;
}
$this->factor[] = $this->num;
return false;
}
} // our class ends here
$example = new Divisors(4567893421);
print $example->countDivisors();
?>