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anonymous

  • 4 years ago

Fool's problem of the day, If \( m \) and \(n\) are positive integers that satisfy \(3m^2 − 8n^2 + 3m^2n^2 = 2008\), then find the product \(m^2n\). Genre: algebra-precalculus Rating: Easy

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  1. asnaseer
    • 4 years ago
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    I have a feeling that we should make use of the fact that \(2008=2^3*251\)

  2. asnaseer
    • 4 years ago
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    I don't know if this helps, but we can rewrite this as:\[3m^2(1+n^2)-2^3n^2=2^3*251\]

  3. asnaseer
    • 4 years ago
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    therefore:\[3m^2(1+n^2)=2^3(251+n^2)\]

  4. anonymous
    • 4 years ago
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    Good point asnaseer.

  5. asnaseer
    • 4 years ago
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    therefore:\[m^2n=\frac{2^3n(251+n^2)}{3(1+n^2)}\]

  6. asnaseer
    • 4 years ago
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    this must be an integer, therefore \(3(1+n^2)\) must divide evenly into \(2^3n(251+n^2)\)

  7. anonymous
    • 4 years ago
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    You are in the right track, divisibility the way :)

  8. asnaseer
    • 4 years ago
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    since \(2^3\) is not divisible by 3, then this implies \(n(251+n^2)\) must be divisible by 3. therefore either n is divisible by 3 and/or \(251+n^2\) is divisible by 3.

  9. anonymous
    • 4 years ago
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    is it 112? (n = 7), makes (251+n^2) =300 which is nice and makes (1+n^2)=50 also nice :D

  10. asnaseer
    • 4 years ago
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    I'm trying to /prove/ the result. I am sure we can use brute force to find /a/ solution, but that doesn't prove it's the /only/ solution. :)

  11. anonymous
    • 4 years ago
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    ah, nvm them, I don't know how to prove it :D

  12. anonymous
    • 4 years ago
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    Yes 112 is the answer :)

  13. asnaseer
    • 4 years ago
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    you win gogind :D

  14. anonymous
    • 4 years ago
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    win by brute force, sounds like cheating :D

  15. anonymous
    • 4 years ago
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    haha, prove it now :D

  16. asnaseer
    • 4 years ago
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    need fooood - I'll be back soon :)

  17. anonymous
    • 4 years ago
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    sure :)

  18. anonymous
    • 4 years ago
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    Good thinking gogind.

  19. asnaseer
    • 4 years ago
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    OK - the furthest I can get with this is as follows:\[\begin{align} 3m^2(1+n^2)-8n^2&=2008\tag{a}\\ m^2n&=\frac{8n(251+n^2)}{3(1+n^2)}\tag{b} \end{align}\]equation (b) implies \(n(251+n^2)\) must be a multiple of 3. 1. lets try n being a multiple of 3.\[\begin{align} n&=3a\\ 3m^2(1+9a^2)-72a^2&=2008\qquad\text{(substituting n=3a into equation(a))}\\ \therefore 3(m^2(1+9a^2)-24a^2)&=2008\\ \qquad&\text{reject since 2008 is not a multiple of 3} \end{align}\] 2. lets try \(251+n^2\) being a multiple of 3.\[\begin{align} 251+n^2&=3a\\ \therefore n^2&=3a-251\tag{c} \end{align}\] Now we can substitute equation (c) into equation (a) to get:\[\begin{align} 3m^2(1+3a-251)-8(3a-251)&=2008\\ 3m^2(3a-250)-24a+\cancel{2008}&=\cancel{2008}\\ 3m^2(3a-250)&=24a\\ \therefore m^2&=\frac{8a}{3a-250}\tag{d} \end{align}\] Equation (d) implies \(a\ge84\). It also shows \(m^2\rightarrow\frac{8}{3}\) as \(a\rightarrow\infty\). From here I could only use trial and error to determine that the only solution to (c) and (d) is a=100, which gives m=4, n=7. Therefore \(m^2n=112\)

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