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anonymous

  • one year ago

Explain why this problem is in a number theory book? I feel like it should be in a Cal 3 book. Prove that if (xₒ,yₒ) is a solution of the linear Diopahntine equation ax - by = 1, then the area of the triangle whose vertices are (0,0), (b,a) and (xₒ,yₒ) is 1/2? This could easily be proven using cross product

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  1. UsukiDoll
    • one year ago
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    The Diophantine equation was in my number theory book as well. An equation which only integer solutions are allowed.

  2. anonymous
    • one year ago
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    eh, probably because it mentions linear Diophantine equations?

  3. anonymous
    • one year ago
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    And there is this problem too. A man pays $1.43 for some apples and pears. If pears cost 17 cents each, and apples, 15 cents each, how many of each did he buy? O.O what the heck? this is 5th grade problem!

  4. UsukiDoll
    • one year ago
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    oh I had that problem too... hey are you reading Number Theory by George Andrews?

  5. anonymous
    • one year ago
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    @UsukiDoll yep :)

  6. anonymous
    • one year ago
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    anyways, this just follows from the determinant: $$\left|\det\begin{bmatrix}b&x_0\\a&y_0\end{bmatrix}\right|=\left|by_0-ax_0\right|=1$$ which is the area of the whose edges are given by \((b,a),(x_0,y_0)\), so the triangle is simply half that: \(1/2\)

  7. UsukiDoll
    • one year ago
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    I've used that book last year in my Elementary Number Theory class XDDDDDDDDDDD!!!!!!!! Luckily I didn't have to do that problem, but I was stunned when I saw that.

  8. UsukiDoll
    • one year ago
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    I've heard that it's the most popular book.

  9. ganeshie8
    • one year ago
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    Mostly the authors choose trivial examples in the start because the focus is on succcessfully applying a particular method, not on getting a quick answer

  10. ganeshie8
    • one year ago
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    the second question translates to solving linear diophantine eqn \[17a+15p = 143\]

  11. anonymous
    • one year ago
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    that problem is fine, it's a Diophantine equation, too: $$15a + 17p = 143\\ 15a=143\pmod{17}\implies -2a=-10\pmod{17}\implies a=5\pmod{17}\\17p=143\pmod{15}\implies 2p=10\pmod{17}\implies p=5\pmod{15}$$

  12. anonymous
    • one year ago
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    oops, the second line is broken

  13. anonymous
    • one year ago
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    $$2p=8\pmod{15}\implies p=8\pmod{15}$$

  14. anonymous
    • one year ago
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    O: I did the second problem by guess and check. Never mind about it being 5th a grade problem ^^

  15. anonymous
    • one year ago
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    I need someone to confirm this problem. Hold on...

  16. anonymous
    • one year ago
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    http://openstudy.com/users/sourwing#/updates/55adfc7be4b045595079d70b

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