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(a) How many ordered pairs $(x,y)$ of integers are there such that $\sqrt{x^2 + y^2} = 5$? Does the question have a geometric interpretation?

(b) How many ordered triples $(x,y,z)$ of integers are there such that $\sqrt{x^2 + y^2 + z^2} = 7$? Does the question have a geometric interpretation?

By geometric interpretation we mean, "Is there a picture that describes the question?"

Guest Mar 29, 2015

Best Answer 

 #2
avatar+80933 
+19

Alan...in the first one, don't (±5, 0) and (0,±5) also "work ??"

And in the second....don't  (±7, 0, 0), (0,±7, 0 ) and (0, 0, ±7) also do the job??

 

   

CPhill  Mar 29, 2015
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3+0 Answers

 #1
avatar+26399 
+10

(a) All combinations of (±3,±4) and (±4,±3) satisfy the equation.  Geometrically, these points lie on the circumference of a circle of radius 5.

 

(b) All combinations of (±2, ±3, ±6), (±2, ±6, ±3), (±6, ±2, ±3), (±6, ±3, ±2),(±3, ±2, ±6) and (±3, ±6, ±2) satisfy the equation.  Geometrically, these points lie on a sphere of radius 7.  See image below:

 Points on sphere

.

Alan  Mar 29, 2015
 #2
avatar+80933 
+19
Best Answer

Alan...in the first one, don't (±5, 0) and (0,±5) also "work ??"

And in the second....don't  (±7, 0, 0), (0,±7, 0 ) and (0, 0, ±7) also do the job??

 

   

CPhill  Mar 29, 2015
 #3
avatar+26399 
+11

Yes, I guess they do.  I forgot about the zeros! 

 

In both cases these points will still lie on the circle of radius 5/sphere of radius 7.

.

Alan  Mar 29, 2015

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