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Difference between revisions of "2005 Alabama ARML TST Problems/Problem 8"

(New page: ==Problem== How many of the positive divisors of 3,240,000 are perfect cubes? ==Solution== {{solution}} ==See Also== *2005 Alabama ARML TST *[[2005 Alabama ARML TST Probl...)
 
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==Problem==
 
==Problem==
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How many of the [[positive]] [[divisor]]s of 3,240,000 are [[perfect cube]]s?
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Find the number of ordered pairs of integers <math>(x,y)</math> which satisfy <center><math>x^2+4xy+y^2=21</math>.</center>
 +
 
 
==Solution==
 
==Solution==
 +
We look at <math>x</math> and <math>y \pmod{3}</math>, since <math>21</math> is a multiple of <math>3</math>.
 +
 +
* Case 1: <math>x\equiv 0\pmod{3}</math>
 +
** Case 1a: <math>y\equiv 0\pmod{3}</math>: Then <math>x^2+4xy+y^2</math> is divisible by <math>3^2=9</math>, but <math>21</math> isn't.
 +
** Case 1b: <math>y\equiv 1\pmod{3}</math>: Then the LHS is <math>1\pmod{3}</math>, while the RHS isn't.
 +
** Case 1c: <math>y\equiv 2\pmod{3}</math>: Then the LHS is <math>1\pmod{3}</math>, while the RHS isn't.
 +
* Case 2: x=1mod3
 +
** Case 2a: y=0mod3: This is equivalent to case 1b.
 +
** Case 2b: y=1mod3: We let <math>x=3x_1+1</math> and <math>y=3y_1+1</math>:
 +
 +
<math>x^2+4xy+y^2=21=(3x_1+1)^2+4(3x_1+1)(3y_1+1)+(3y_1+1)^2=9(x^2+y^2+4x_1y_1+2x_1+2y_1)+6</math>
 +
 +
But 21 isn't 6mod9, it's 3mod9.
 +
 +
** Case 2c: y=2mod3: Then the LHS is 1mod3 while the RHS isn't.
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* Case 3: x=2mod3
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** Case 3a: y=0mod3: This is equivalent to case 1c.
 +
** Case 3b: y=1mod3: This is equivalent to case 2c.
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** Case 3c: y=2mod3: We let <math>x=3x_1+2</math> and <math>y=3y_1+2</math>:
 +
 +
<math>x^2+4xy+y^2=21=(3x_1+2)^2+4(3x_1+2)(3y_1+2)+(3y_1+2)^2=9(x_1^2+y_1^2+4x_1y_1+4x_1+4y_1+2)+6</math>
 +
 +
But 21 isn't 6mod9, it's 3mod9.
  
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{{solution}}
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Therefore, there are absolutely no solutions to the above equation.
  
 
==See Also==
 
==See Also==
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*[[2005 Alabama ARML TST]]
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{{ARML box|year=2005|state=Alabama|num-b=7|num-a=9}}
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*[[2005 Alabama ARML TST Problems/Problem 7 | Previous Problem]]
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−
*[[2005 Alabama ARML TST Problems/Problem 9 | Next Problem]]
 
  
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[[Category:Introductory Number Theory Problems]]
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[[Category:Intermediate Number Theory Problems]]

Latest revision as of 00:38, 28 October 2015

Problem

Find the number of ordered pairs of integers $(x,y)$ which satisfy

$x^2+4xy+y^2=21$.

Solution

We look at $x$ and $y \pmod{3}$, since $21$ is a multiple of $3$.

  • Case 1: $x\equiv 0\pmod{3}$
    • Case 1a: $y\equiv 0\pmod{3}$: Then $x^2+4xy+y^2$ is divisible by $3^2=9$, but $21$ isn't.
    • Case 1b: $y\equiv 1\pmod{3}$: Then the LHS is $1\pmod{3}$, while the RHS isn't.
    • Case 1c: $y\equiv 2\pmod{3}$: Then the LHS is $1\pmod{3}$, while the RHS isn't.
  • Case 2: x=1mod3
    • Case 2a: y=0mod3: This is equivalent to case 1b.
    • Case 2b: y=1mod3: We let $x=3x_1+1$ and $y=3y_1+1$:

$x^2+4xy+y^2=21=(3x_1+1)^2+4(3x_1+1)(3y_1+1)+(3y_1+1)^2=9(x^2+y^2+4x_1y_1+2x_1+2y_1)+6$

But 21 isn't 6mod9, it's 3mod9.

    • Case 2c: y=2mod3: Then the LHS is 1mod3 while the RHS isn't.
  • Case 3: x=2mod3
    • Case 3a: y=0mod3: This is equivalent to case 1c.
    • Case 3b: y=1mod3: This is equivalent to case 2c.
    • Case 3c: y=2mod3: We let $x=3x_1+2$ and $y=3y_1+2$:

$x^2+4xy+y^2=21=(3x_1+2)^2+4(3x_1+2)(3y_1+2)+(3y_1+2)^2=9(x_1^2+y_1^2+4x_1y_1+4x_1+4y_1+2)+6$

But 21 isn't 6mod9, it's 3mod9.

Therefore, there are absolutely no solutions to the above equation.

See Also

2005 Alabama ARML TST (Problems)
Preceded by:
Problem 7 		Followed by:
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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