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Generating Z^2
Let $a=(a_1,a_2)\in\mathbb Z^2$ where $a_1$ and $a_2$ are coprime. Show that there exists $b=(b_1,b_2)\in\mathbb Z^2$ such that any element of $\mathbb Z^2$ can be written as a $\mathbb Z$-linear combination of $a$ and $b$.
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"Everything should be made as simple as possible, but not simpler." - Albert Einstein