Positive divisors

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January 25th 2010, 06:18 AM
davidman

Positive divisors

There are $x$ number of positive divisors of $72$ , $y$ number of positive divisors of $900$ .

The positive divisors of $900$ that are not divisors of $72$ are $z$ .

Hoping there's a formula and you're not actually supposed to go over all possible numbers... (Giggle)

Any help appreciated.
January 25th 2010, 06:40 AM
Plato

Quote:

Originally Posted by davidman

There are $x$ number of positive divisors of $72$ , $y$ number of positive divisors of $900$ .
The positive divisors of $900$ that are not divisors of $72$ are $z$ .

I find this question a bit confusing. Is this what you mean?
Suppose that $t$ is the number of divisors of $72~\&~900$ .
Then $z=y-t$ . Note that $t\le x$ .

If that is not your question, please clarify in detail.
January 25th 2010, 07:51 AM
davidman

Not quite sure what your $t$ represents.

$z$ is the number of divisors of $900$ minus the divisors that overlap with $72$ .
January 25th 2010, 08:09 AM
Plato

Quote:

Originally Posted by davidman

Not quite sure what your $t$ represents.
$z$ is the number of divisors of $900$ minus the divisors that overlap with $72$ .

What is the point of this reply?
You did not read my reply? It says clearly what $t$ equals.
Do you have difficulty translating Enghish?
January 25th 2010, 08:15 AM
davidman

Quote:

Originally Posted by Plato

What is the point of this reply?

First, to tell you that I was not quite sure what you meant with what you defined $t$ to be. In other words, $72~\&~900$ is confusing and threw me off, so I do not understand what you mean when you say "the number of divisors of $72~\&~900$ ". Is it the number of divisors that overlap for the two? Is it the sum of the number of divisors, not taking into regard the case of overlap?

Second, to clarify what it was I meant. You did ask me to, didn't you?

Quote:

You did not read my reply? It says clearly what $t$ equals.
Do you have difficulty translating Enghish?

English is my first language. I just don't have a very wide vocabulary when it comes to mathematics.

But sure, I'll look over your post again for hints to what I might not be seeing clearly.
January 25th 2010, 08:37 AM
Plato

What does it mean to say that a number is a divisor of seventy-two and ninety?
Does it mean that the number divides $72$ and divides $90$ ?
BTW. ‘Overlap’ in not mathematical.
January 25th 2010, 09:07 AM
davidman

Quote:

Originally Posted by Plato

What does it mean to say that a number is a divisor of seventy-two and ninety?
Does it mean that the number divides $72$ and divides $90$ ?

Yes, that makes sense. If we assume that is what it means, how do you figure out the number of divisors of a certain number (or two numbers for that matter)? Would be great to know how.
January 25th 2010, 10:33 AM
Plato

Any positive integer can be factors as powers of primes.
$900=2^2\cdot 3^2\cdot 5^2$ look at the exponents:
Then add one to each exponent and multiply: $(2+1)(2+1)(2+1)=27$ .
So there are 27 divisors of 900.

$72=2^3\cdot 3^2$ , so $(3+1)(2+1)=12$ .
There 12 divisors of 72.

The greatest common divisor: $\text{GCD}(900,72)=2^2\cdot 3^2$ so $(2+1)(2+1)=9$ common divisors of both 72 and 900.

Thus there are $27-9=18$ divisors of 900 that are not divisors of 72.
January 26th 2010, 05:36 AM
davidman

Thank you so much! It was a lot more straightforward than I expected. (Surprised)