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amistre64

find 2 integers such that they sum to: 1640 and their LCM is: 8400

  • one year ago
  • one year ago

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  1. amistre64
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    my idea is to factor the LCM to create a pool of options

    • one year ago
  2. amistre64
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    8400 = 84*100 = 84*2*2*5*5 = 2*2*3*7*2*2*5*5 2*2*2*2*3*5*5*7, therefore 2 2*2 2*2*2 2*2*2*2 2*2*2*2*3 2*2*2*2*3*5 2*2*2*2*3*5*5 2*2*2*2*3*5*5*7 etc ... but that does seem like a long way around it

    • one year ago
  3. CliffSedge
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    Maybe algebra with a quadratic equation?

    • one year ago
  4. amistre64
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    maybe. but number theory methods might be perfered

    • one year ago
  5. CliffSedge
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    Might want to use optimization methods from calculus too to minimize the coefficients. Yes, it does seem like a number theory issue, but algebra is always my starting point for solving unknowns.

    • one year ago
  6. PaxPolaris
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    \[\large x \left( 1640-x \right)=8400n\] where n is a natural number

    • one year ago
  7. amistre64
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    hmm, the "n" seems interesting

    • one year ago
  8. amistre64
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    on the test i just ended up using the y= on my ti83 y1 = 1640-x y2= lcm(x,1640-x) then searched thru that table till i found y2 = 8400

    • one year ago
  9. PaxPolaris
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    \[x=820 \pm \sqrt {820^2-8400n}\]

    • one year ago
  10. amistre64
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    i assume the under radical needs to remain 0 or greater?

    • one year ago
  11. PaxPolaris
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    needs to be a perfect square

    • one year ago
  12. amistre64
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    lol, yeah i spose that would have to be a major caveat seeing the x needs to be an integer :)

    • one year ago
  13. PaxPolaris
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    \[\large x=820 \pm 400\sqrt{1681-21n}\]

    • one year ago
  14. amistre64
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    i think one more condition was such that x and y were both positive values

    • one year ago
  15. PaxPolaris
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    sorry, \[\large x=820 \pm \sqrt{400(1681-21n)}\]\[\large x=820 \pm 20\sqrt{1681-21n}\] and n is the GCF of x and y

    • one year ago
  16. robtobey
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    240 + 1400 = 1640, LCM[240, 1400] = 8400 Used the following small program and fed it to Mathematica. Table[{n, LCM[n, 1640 - n] == 8400}, {n, 820}] Part of the Output: {233, False}, {234, False}, {235, False}, {236, False}, {237, False}, \ {238, False}, {239, False}, {240, True}, {241, False}, {242, False}, \ {243, False}, {244, False},

    • one year ago
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