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please write a proof: There exists a minimum value of k ∈ N such that for...

please write a proof:

There exists a minimum value of k ∈ N such that for every positive integer n ≥ k, ∃x, y ∈ N ∪ {0} such that n = 4x + 7y.

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Answer #1

We will prove that for any , there exist such that .

So for it can be written as of the form for some .

Now for any natural numbers which is greater than 16 can be written as of the forn or .

As   it can be written as of the form for some so   and   also can be written as of the form   for some .

Hence there exist a minimum value such that for every positive integer , such that .

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