The Great Depression is a period of time of American depression.

Let \(a,b \in \mathbb{Z}\), not both zero. \(\gcd (a,b)\) is the greatest value \(d\) such that \(d|a\), \(d|b\).

greatest common divisor is a linear combination

We can write \(\gcd (a,b) = as+bt\) for some \(s,t \in \mathbb{Z}\).

Let us define:

\begin{equation} S = \{am + bn: m,n \in \mathbb{Z}, am+bn > 0\} \end{equation}

We will first check that \(S\) is non-empty. To do so, let \(a\) be negative and \(b\) be positive. Then, set \(m = -1\), \(n = 1\). We can see that \(am + bn > 0\), satisfying the conditions of the set. In a similar manner, we can demonstrate that regardless of the choice of \(a, b\), \(S\) is non-empty.

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