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1. If $a$ and $b$ must be nonnegative integers, what is the largest integer $n$ such that $13a + 18b = n$ has no solutions?

2. If $a$ and $b$ must be positive integers, what is the largest integer $n$ such that $13a + 18b = n$ has no solutions?

 Jun 30, 2021
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Note: If you are wondering how to prove the Frobenius Coin Theorem, it is trivial by Bezout's Lemma. Induction also comes in handy, though I would not know how you would prove it using induction. Proof by contradiction generally is a solid and reliable approach, but I do not think it comes in handy on this problem.

 

1. By the Frobenius Coin Theorem, the answer is $\text{lcm}(13, 18) - (13 + 18)$

Because they are relatively prime, the lcm is their product, namely $13 \cdot 18 = 234.$

Thus the answer is $234 - 31 = \boxed{204}$

 

I will leave 2 up to you.

 Jun 30, 2021
edited by MathProblemSolver101  Jun 30, 2021

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