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AMC 8 Math Olympiad

Integer | ISI-B.stat Entrance(Objective from TOMATO) | Problem 72

Try this beautiful problem from Integer from TOMATO useful for ISI B.Stat Entrance. You may use sequential hints to solve the problem.

Try this beautiful problem Based on Integer useful for ISI B.Stat Entrance.

Integer| ISI B.Stat Entrance | Problem 72


The number of integer (positive ,negative or zero)solutions of \(xy-6(x+y)=0\) with \(x\leq y\) is

  • 5
  • 10
  • 12
  • 9

Key Concepts


Integer

Algebra

Divisor

Check the Answer


Answer: 10

TOMATO, Problem 72

Challenges and Thrills in Pre College Mathematics

Try with Hints


Factorize the given equation

Can you now finish the problem ……….

Find the divisor

can you finish the problem……..

Given equation is \(xy-6(x+y)=0\)

\(\Rightarrow xy-6x-6y=0\)

\(\Rightarrow xy-6x-6y+36=36\)

\(\Rightarrow (x-6)(y-6)=3^2 \times 2^2\)

Now the numbers of factpr of \(36=9\) i.e \(\{1,2,3,4,6,9,12,18,36\}\)

Thus we may say that 36 has 9 positive divisors, and 9 negative. and x=0 and y=0 is also a solution

the given condition \(x\leq y\) ,so there are 10 non-negetive solution

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