Find the expression...

**jzellt** · February 16th 2010, 08:17 PM

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.

**icemanfan** · February 16th 2010, 08:24 PM

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.

**jzellt** · February 16th 2010, 08:41 PM

Ok, it won't be a problem for me to find those values...
But how does that lead to my desired expression?

**icemanfan** · February 16th 2010, 08:55 PM

Originally Posted by jzellt

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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