Difference between revisions of "2000 AIME II Problems/Problem 2"
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== Problem == | == Problem == | ||
| − | A point whose coordinates are both integers is called a lattice point. | + | A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math>x^2 - y^2 = 2000^2</math>? |
== Solution == | == Solution == | ||
| − | { | + | <cmath>(x-y)(x+y)=2000^2=2^8 \cdot 5^6</cmath> |
| + | |||
| + | Note that <math>(x-y)</math> and <math>(x+y)</math> have the same [[parity|parities]], so both must be even. We first give a factor of <math>2</math> to both <math>(x-y)</math> and <math>(x+y)</math>. We have <math>2^6 \cdot 5^6</math> left. Since there are <math>7 \cdot 7=49</math> factors of <math>2^6 \cdot 5^6</math>, and since both <math>x</math> and <math>y</math> can be negative, this gives us <math>49\cdot2=\boxed{098}</math> lattice points. | ||
== See also == | == See also == | ||
{{AIME box|year=2000|n=II|num-b=1|num-a=3}} | {{AIME box|year=2000|n=II|num-b=1|num-a=3}} | ||
| + | |||
| + | [[Category:Intermediate Number Theory Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 19:30, 4 July 2013
Problem
A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola 
?
Solution
![\[(x-y)(x+y)=2000^2=2^8 \cdot 5^6\]](http://latex.artofproblemsolving.com/2/b/8/2b87b01628eb920d6fd80ecee4d30daf02966520.png)
Note that 
and 
have the same parities, so both must be even. We first give a factor of 
to both and . We have 
left. Since there are 
factors of , and since both 
and 
can be negative, this gives us 
lattice points.
See also
| 2000 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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