1. ## 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...

2. Two hints:

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

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

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

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

2. There are an infinite number of possible answers.

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

4. 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 $\displaystyle a, b, c$ such that

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

Now write

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

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