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**gyanshrestha****Member**- Registered: 2007-11-06
- Posts: 41

three angles of a triangle are 5x+3y, 10y+30 and 3x+20

what is the value of x+y

any idea ?

this question is asked on a mathematical competition.

http://gyan.talkacademy.com.np

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

three angles of a triangle are 5x+3y, 10y+30 and 3x+20

what is the value of x+y

You left out the condition that x,y have to be positive integers. x + y = 15 is the answer.

You need to solve the linear diophantine equation

There are several ways to do this, continued fractions, Brahmagupta's method, computer solution or trial and error.

In the case of this small problem, trial and error is fastest. If you are skilled in a programming language you can write a routine to do it quickly.

Once you have found the answer you can check its uniqueness by expanding the generating function.

When expanded you will see that the coefficient of x^130 is 1 proving there is only one solution.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Actually, there is an easier method for this particular problem.We can note that x must be divisible by 13. Since it has to be positive, it cannot be 0, and since 26 is too big, it must be 13. The value of y follows immediately.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

I did mention trial and error and naturally I would start with the y's first because they have the biggest coefficient.

Of course until you follow what the guy in post #2 said you will not get to that diophantine equation.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Actually, I will. Don't you think I know how to get to the Diophantine?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

oooga ooooga chaka oooga chaka.

Post #2, which essentially solves the problem, provides the answer and still leaves some mystery for the OP was not meant for you.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

I do not consider post #3 trial and error.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

I did not say it was exactly but it does sound like it. What I am saying is that the trail and error approach recommended by the dude in post #2 would get the answer just as quickly.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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