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Use the Well Ordering Principle to prove that any integer greater than or equal to 23 can be represented as the sum of nonnegative integer multiples of 6, 7 and 17.

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Let S(n) be the proposition that n can be represented as the sum of nonnegative integer multiples of 6, 7 and 17.

Let C=\{n\geq 23\ |\ S(n) \ \text{is false}\ \} .

We assume that C is nonempty.

Then by the Well Ordering Principle, there is a smallest number, m\in C .


23=17+6\\ 24=6\cdot 4 \\ 25=6\cdot 3+7\\ 26=6\cdot 2+7\cdot 2 \\ 27=6+7\cdot 3 \\28=7\cdot 4


Therefore, m\geq29 .

Since m>m-6\geq 23 , it follows that (m-6)\not\in C and S(m-6) is true.

But if S(m-6) is true, then S(m) must be true. Contradiction.

(if n can be represented as the sum of nonnegative integer multiples of 6, 7 and 17, then we can add 6 and get representation for number n+6 )


So, C is empty and S(n) is true for n\geq 23 .

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Question ID: mtid-5-stid-8-sqid-1146-qpid-884