How do I prove that if $n≥12$, then $n=3a+7b$ for some $a≥0$, $b≥0$ where $a,b ∈ N$?

Question

How do I prove that if $n\geq 12$, then $n=3a+7b$ for some $a\geq 0$, $b\geq 0$ where $a,b \in \mathbb{N}$?

I'm having trouble in solving this algebraic type of proof. Your help is greatly appreciated.

Answer 1

it is enough to give such expressions for $n = 12, 13, 14.$ Any larger $n$ can then be given by simply increasing the $a$ value from one of those expressions.

$$ 6 + 7 = 13 $$

make a separate check for smaller $n$

Answer 2

This is basically the chicken mcnugget theorem.

It states that if $a$ and $b$ are coprime integers then the largest number that cannot be expressed as $as+bt$ with $s,t\in \mathbb N$ is $ab-a-b$.

Proof:

Every integer $n$ is expressable in the form $as+tb$ such that $s\in \{0,1,2,\dots, b-1\}$ and $t\in \mathbb Z$. To see this notice that $as$ only has to have the same value as $n\bmod b$ so that we can use $t$ to make $as+tb=n$. This is possible because $\{0,1,\dots, b-1\}$ is a full residue class $\bmod b$ and $a$ is coprime to $b$.

It is now clear that the biggest number of this form such that $t$ is negative is $(b-1)a-1(b)=ab-a-b$.