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### Every $n > 17$ is a non-negative integer combination of $4$ and $7$. [duplicate]

Prove that every integer greater than $17$ is a non-negative integer combination of $4$ and $7$. In other words, for all natural numbers $n$, $n$ greater or equal to $17$, there exists non-negative ...
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### For coprime $p$ and $q$, what is the largest number $m$ that can not be written as $ap+bq$ for $a,b \geq 0$.
This is a sub-problem that I am solving in light of a larger more general problem. I believe it is probably quite easy but I often struggle with number theory. Here is my theorem: the number $m$ is ...