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Math Help - Find the expression...

  1. #1
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    Find the expression...

    Show that if a1,...,ak are non-zero integers, then their GCD has the form a1u1+...+akuk for some integers u1,...,uk. Find such an expression where k=3 and a1=1092, a2=1155, and a3=2002.

    Any advice... I don't know where to even begin...

    Thanks in advance.
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  2. #2
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    Two hints:

    1. Find values u_1, u_2, v_1, v_2, w_1, w_2 for all three of the following:

    1092u_1 + 1155u_2 = \gcd (1092, 1155)

    1092v_1 + 2002v_2 = \gcd (1092, 2002)

    1155w_1 + 2002w_2 = \gcd (1155, 2002)

    2. There are an infinite number of possible answers.
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  3. #3
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    Ok, it won't be a problem for me to find those values...
    But how does that lead to my desired expression?
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  4. #4
    MHF Contributor
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    Quote Originally Posted by jzellt View Post
    Ok, it won't be a problem for me to find those values...
    But how does that lead to my desired expression?
    Easily. There are integers a, b, c such that

    a \cdot \gcd (1092, 1155) + b \cdot \gcd (1092, 2002) + c \cdot \gcd (1155, 2002) = \gcd (1092, 1155, 2002).

    Now write

    a(1092u_1 + 1155u_2) + b(1092v_1 + 2002v_2) + c(1155w_1 + 2002w_2) = \gcd (1092, 1155, 2002), or equivalently:

    1092(au_1 + bv_1) + 1155(au_2 + cw_1) + 2002(bv_2 + cw_2) = \gcd (1092, 1155, 2002)
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