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Math Help - linear combination ?

  1. #1
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    linear combination ?

    I have a question that involves two parts. I believe I have the first part correct. The second part is what I am confused on and explain below.

    a) Find the gcd(260,154)

    260 = 1(154) + 106
    154 = 1(106) + 48
    106 = 2(48) +10
    48 = 4(10) + 8
    10 = 1(8) + 2
    8 = 4(2) + 0

    Thus the gcd(260,154) is 2 since it is the last non-zero remainder.

    ***b) Find integers such that 154x + 260y = 4

    Well, usually I would use part a and find the line with 4 as a remainder and work backwards until I find x and y. My problem is that I don’t have 4 as a remainder in any of the lines above and therefore I am not sure what to do? I may have calculated the gcd incorrectly, but I have checked it several times and get the same answer. Also, my book examples find x and y differently than the way I am being taught. Any suggestions? Thanks!
    Last edited by MathStudent1; March 14th 2007 at 03:00 PM.
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  2. #2
    Junior Member frenzy's Avatar
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    Quote Originally Posted by MathStudent1 View Post

    ***b) Find integers such that 154x + 240y = 4
    240=154(1)+86
    154=86(1)+68
    86=68(1)+18
    18=14(1)+4
    14=4(3)+2
    4=2(2)

    ---------

    2=14-4(3)
    2=(68-18(3))-(18-14)(3)
    2=68-18(6)+14(3)
    2=(154-86)-(86-68)(6)+(68-18(3))(3)
    2=154-86(7)+68(9)-18(9)
    2=154-(240-154)(7)+(154-86)(9)-(86-68)(9)
    2=-240(7)+154(17)-86(18)+68(9)
    2=-240(7)+154(7)-(240-154)(18)+(154-86)(9)
    2=-240(25)+154(44)-(240-154)(9)
    2=-240(34)+154(53)
    2=240(-34)+154(53)

    240(-68)+154(106)=4

    can you see what i did?
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  3. #3
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    Quote Originally Posted by frenzy View Post
    240=154(1)+86
    154=86(1)+68
    86=68(1)+18
    18=14(1)+4
    14=4(3)+2
    4=2(2)

    ---------

    2=14-4(3)
    2=(68-18(3))-(18-14)(3)
    2=68-18(6)+14(3)
    2=(154-86)-(86-68)(6)+(68-18(3))(3)
    2=154-86(7)+68(9)-18(9)
    2=154-(240-154)(7)+(154-86)(9)-(86-68)(9)
    2=-240(7)+154(17)-86(18)+68(9)
    2=-240(7)+154(7)-(240-154)(18)+(154-86)(9)
    2=-240(25)+154(44)-(240-154)(9)
    2=-240(34)+154(53)
    2=240(-34)+154(53)

    240(-68)+154(106)=4

    can you see what i did?
    Frenzy, I made a TYPO!! I wrote part a correct and messed up part b. It should read 154x + 260y = 4. So, how would you do it using 260 instead of 240? Thanks!
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  4. #4
    Junior Member frenzy's Avatar
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    Quote Originally Posted by MathStudent1 View Post
    Frenzy, I made a TYPO!! I wrote part a correct and messed up part b. It should read 154x + 260y = 4. So, how would you do it using 260 instead of 240? Thanks!

    2=10-8
    2=(106-48(2))-(48-10(4))
    2=106-48(3)+10(4)
    2=(260-154)-(154-106)(3)+(106-48(2))(4)
    2=260-154(4)+106(7)-48(8)
    2=260-154(4)+(260-154)(7)-(154-106)(8)
    2=260(8)-154(19)+106(8)
    2=260(8)-154(19)+(260-154)(8)
    2=260(16)-154(27)
    2=260(16)+154(-27)
    4=260(32)+154(-54)

    How's that?
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  5. #5
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    Just to clarify my thinking. In the last line when you multiply through by 2 to both sides on the RHS you multiply 16 and -27 by two and not 260 and 154 to achieve the desired linear combination. I think I get it. Thanks!
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  6. #6
    Junior Member frenzy's Avatar
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    Quote Originally Posted by MathStudent1 View Post
    Just clarify one thing. In the last line when you multiply through by 2 to both sides one the RHS you multiply 16 and -27 by two and not 260 and 154 to achieve the desired linear combination.

    Correct
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