Difference between revisions of "1969 AHSME Problems/Problem 19"

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== Solution ==
 
== Solution ==
Let <math>(xy)^2=a</math>. The expression given is equal to <math>a^2-10a+9=0</math>, which can be factored as <math>(a-9)(a-1)=0</math>. Because <math>x</math> and <math>y</math> are positive, we can eliminate the possibilities where <math>a</math> is negative. Thus, we have <math>a=(xy)^2=9</math> and <math>a=(xy)^2=1</math>. From here it is easy to see that the only integral pairs of <math>x</math> and <math>y</math> are <math>(3, 1), (1, 3)</math>, and <math>(1, 1)</math>. The answer is <math>\fbox{B}</math>.
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Let <math>(xy)^2=a</math>. The expression given is equal to <math>a^2-10a+9=0</math>, which can be factored as <math>(a-9)(a-1)=0</math>. Thus, we have <math>a=(xy)^2=9</math> and <math>a=(xy)^2=1</math>. Because <math>x</math> and <math>y</math> are positive, we can eliminate the possibilities where <math>xy</math> is negative. From here it is easy to see that the only integral pairs of <math>x</math> and <math>y</math> are <math>(3, 1), (1, 3)</math>, and <math>(1, 1)</math>. The answer is <math>\fbox{B}</math>.
  
 
== See also ==
 
== See also ==

Latest revision as of 16:14, 10 July 2015

Problem

The number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4-10x^2y^2+9=0$ is:

$\text{(A) } 0\quad \text{(B) } 3\quad \text{(C) } 4\quad \text{(D) } 12\quad \text{(E) } \infty$

Solution

Let $(xy)^2=a$. The expression given is equal to $a^2-10a+9=0$, which can be factored as $(a-9)(a-1)=0$. Thus, we have $a=(xy)^2=9$ and $a=(xy)^2=1$. Because $x$ and $y$ are positive, we can eliminate the possibilities where $xy$ is negative. From here it is easy to see that the only integral pairs of $x$ and $y$ are $(3, 1), (1, 3)$, and $(1, 1)$. The answer is $\fbox{B}$.

See also

1969 AHSC (ProblemsAnswer KeyResources)
Preceded by
Problem 18
Followed by
Problem 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
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