NMTC Problem

Number Theory Level 3

Find the number of natural number pairs $(x,y)$ in which $x>y$ and $\frac{5}{x}+\frac{6}{y}=1$ .

The answer is 3.

2 solutions

Akshat Sharda
Aug 23, 2015

$\Rightarrow \frac{5}{x}+\frac{6} {y}=1$

$\Rightarrow 5y+6x=xy$

$\Rightarrow xy-5y-6x=0$

$\Rightarrow y(x-5)-6x=0$

Adding $\color{#D61F06}{30}$ both sides so that we can factorize and solve the problem further ,

$\Rightarrow y(x-5)-6x+\color{#D61F06}{30}=\color{#D61F06}{30}$

$\Rightarrow y(x-5)-6(x-5)=\color{#D61F06}{30}$

$\Rightarrow (x-5)(y-6)=\color{#D61F06}{30}$

Now , $\color{#D61F06}{30} \Rightarrow (30×1),(15×2),(10×3)$ and $(6×5)$ .

Since $x>y$ , $(x,y) \Rightarrow (35,7),(20,8)$ and $(15,9)$ .

Thus , only $\huge \boxed{3}$ such pairs are possible.

Was 30 used because it was the lowest common factor?

Nehemiah Osei - 5 years, 9 months ago

I added $\color{#D61F06}{30}$ so as to factorize it and simply get the results as by factorizing in such problem can be a great help.

^_^

Akshat Sharda - 5 years, 9 months ago

Sorry, I meant multiple.

I would really like to use this approach to solve some problems but I want to find out if your choice of 30 was because it was the lowest common multiple.

There may be some questions of this nature that one may not easily tell what should be added.

So the actual question is, is there any special method used to get the 30? If so was it by finding the lowest common multiple of the two numbers

Nehemiah Osei - 5 years, 9 months ago

@Nehemiah Osei – Ok. Now I got you. Yes , you are correct , as $\color{#D61F06}{30}$ is the $LCM$ we added it to complete factors.

You can try a similar problem here .

You can also get some help from here .

Akshat Sharda - 5 years, 9 months ago

@Akshat Sharda – Thanks. The links were useful

Nehemiah Osei - 5 years, 9 months ago

@Nehemiah Osei – Ok $\ddot \smile$ ......

Akshat Sharda - 5 years, 9 months ago

An NMTC 2015 problem right????????????

Ankit Kumar Jain - 5 years, 9 months ago

Yeah.......Look at the topic $\ddot \smile$ $\huge 2015$

Akshat Sharda - 5 years, 9 months ago

Neeraj Kamal
Sep 13, 2015

Here x and y are natural no. also x>y , Now A.T.Q 5/x + 6/y = 1 So, 5/x = y-6/y , now we know that x>y so 5 also should be greater than y-6 that means 11>y, also y cannot be smaller than 6 because that results in y>x now y lies between 6 and 11 and also x and y are natural no. So the value of y that gives integer values of x are 7,8,9 so there are 3 pairs.

NMTC Problem

The answer is 3.

2 solutions

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