Exercise 1.2.24 (No integers $x,y$ exist satisfying $x + y = 100$ and $(x,y) = 3$)

Question

Prove that no integers $x,y$ exist satisfying $x + y = 100$ and $(x,y) = 3$.

Richard Ganaye · Accepted Answer

\begin{proof} Assume for contradiction that $x + y = 100$ and $(x,y) = 3$. Then $3 \mid x$ and $3 \mid y$, so $3 \mid x + y = 100$.  But $3 
mid 100$. This contradiction shows that no integers $x,y$ exist satisfying $x + y = 100$ and $(x,y) = 3$.
\end{proof}

Exercise 1.2.24 (No integers $x,y$ exist satisfying $x + y = 100$ and $(x,y) = 3$)

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Exercise 1.2.24 (No integers $x,y$ exist satisfying $x + y = 100$ and $(x,y) = 3$)

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