# Number of solutions

1. May 6, 2013

### Saitama

1. The problem statement, all variables and given/known data
Consider the equation $\frac{xy}{x+y}=2^3\times 3^4 \times 5^6$ then the number of positive integral solutions of equation are
A)140
B)819
C)72
D)None

2. Relevant equations

3. The attempt at a solution
Honestly, I am out of ideas on this one. I don't even have a clue about how should I start. -_-

Any help is appreciated. Thanks!

2. May 6, 2013

### Staff: Mentor

Hint: if (a,b) with x!=y is a solution for (x,y), what about (b,a)?

Based on that, is the number of solutions even or odd?

3. May 7, 2013

### Saitama

Are you sure about the factorial sign or is it a typo? -.-'

4. May 7, 2013

### Staff: Mentor

I believe mfb is using a common programming symbol ❲viz., !=❳ that equates to
the mathematician's sign.

5. May 7, 2013

### Saitama

Ah yes, now I recognise it, thanks!

I am confused about mfb's first question. If (a,b) is a solution when x!=y, then (b,a) is surely a solution or am I missing something?

6. May 7, 2013

### Staff: Mentor

No, no, it sounds as though you do understand what he is saying there.

7. May 7, 2013

### haruspex

Yes, that's exactly mfb's point. So are there an odd or even number of such solutions? What about solutions with x=y?

But to me this is a somewhat unsatisfactory way of answering the question. It would be more educational to come up with the number as if it were not multiple choice. The trick is not to be put off by the huge number on the right. If you were given xy/(x+y) = a, how might you simplify that?

8. May 7, 2013

### Saitama

Equal number of odd and even solutions?

With x=y, there is only one solution.

$xy=xa+ya \Rightarrow xy-xa-ya=0 \Rightarrow xy-xa+a^2-ya=a^2$
$\Rightarrow x(y-a)-a(y-a)=a^2 \Rightarrow (x-a)(y-a)=a^2$

Does this look good?

9. May 7, 2013

### Dick

Yes, that's great! The best thing you can do with an integer problem is factor it. Now the next step is to figure out the number of ways to factor (2^3*3^4*5^6)^2, right?

10. May 7, 2013

### Saitama

Do I need to use the Divisor function?

11. May 7, 2013

### Dick

12. May 7, 2013

### Saitama

Okay so that gives me a total of (6+1)(8+1)(12+1)=7*9*13=819 solutions.

Thank you everyone!

13. May 7, 2013

### haruspex

No, it's much simpler than that. How many possible values are there for u = x-a given that uv = 2638512?

14. May 7, 2013

### Dick

That IS the divisor function, isn't it?

15. May 8, 2013

### haruspex

Sorry, I was thinking of something else.

16. May 8, 2013

### Staff: Mentor

I gave the original equation to wolfram alpha, but it spins its wheels without making progress. I think the task must be exceeding the slice of time it allocates to a freeloader.

17. May 8, 2013

### Staff: Mentor

Well, it will not work for arbitrary given options, but it is the first solution I found and the "proper" solution takes more time.