No Integer Solutions 5C4107

1. **State the problem:** We want to prove that there are no integers $a$ and $b$ such that the equation $$12a - 28b = 11$$ holds. 2. **Recall the key concept:** For an equation of the form $$Ax + By = C$$ to have integer solutions $x$, $y$, the greatest common divisor (gcd) of $A$ and $B$ must divide $C$. 3. **Calculate the gcd:** $$\gcd(12, 28) = 4$$ 4. **Check divisibility:** Since $4$ divides $12$ and $28$, for integer solutions to exist, $4$ must divide $11$. 5. **Evaluate divisibility:** $11$ divided by $4$ leaves a remainder, so $4 \nmid 11$. 6. **Conclusion:** Because the gcd of $12$ and $28$ does not divide $11$, there are no integers $a$ and $b$ satisfying $$12a - 28b = 11$$. This completes the proof.