Try this beautiful problem from Algebra based on Largest Common Divisor .
For natural numbers
But try the problem first...
Answer:$3$
PRMO-2018, Problem 21
Pre College Mathematics
First hint
At first we have to find out the divisors that satisfy the equation \( xy=x+y+(x,y)\) .so we assume that \(x\)=ak and \(y\)=bk and try to find out the divisors
Can you now finish the problem ....
Second Hint
Let \(x\) =ak and \(y\)=bk, then (x,y)=k and (a,b)=1
Therefore \(xy=x+y+(x,y)\)
\(\Rightarrow abk^2=ka+kb+k\)
\(\Rightarrow kab=a+b+1\)
Can you finish the problem...
Final Step
\(k=1\Rightarrow ab=a+b+1\Rightarrow a=1=\frac{2}{b-1}\)
For \(a\in \mathbb N\), then (b-1) divides 2
\(\Rightarrow b-1=1,2\)
\(\Rightarrow b=2,3\)
Therefore a=1+2=3 and a=1+1=2
\(\Rightarrow (x,y)=(3,2) or (2,3)\)
Now for \(x=y\Rightarrow(x,y)=x\)
so \(xy=x+y(x<y)\)
\(\Rightarrow x^2=3x\)
\(\Rightarrow x^2 -3x=0\)
\(\Rightarrow x(x-3)=0\)
\(\Rightarrow x=3 or 0\)
Therefore \(x\in \mathbb N\Rightarrow x=3,y=3\)
Therefore \((x,y)=(3,3)\)
Hence total number of pairs =3
From the video below, let's learn from Dr. Ashani Dasgupta (a Ph.D. in Mathematics from the University of Milwaukee-Wisconsin and Founder-Faculty of Cheenta) how you can shape your career in Mathematics and pursue it after 12th in India and Abroad. These are some of the key questions that we are discussing here:
