So I have a question that I got stuck on, it says I have to prove that all integers greater than 17 can be written using a sum of 7's and 4's. For instance 7 + 7 + 7 + 4 + 4 = 29 or 7 + 4 + 4 + 4 = 19 .... etc. Appreciate the help. Cheers.
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2FYI, this is a specific case of the coin problem. – John Omielan Apr 21 '20 at 01:13
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1What have you done on it so far? – saulspatz Apr 21 '20 at 01:17
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If $x$ has atleast one $7$, remove it and add two $4$'s. If $x$ has all $4$'s remove five $4$'s and add three $7$'s. (In the latter case $x \geq 20$ necessarily).
If you need to use strong induction, then we have for $x-4$ when $x \geq 22$, that $x$ satisfies the condition. Then the cases for $18,19,20,21$ are clear.
Derek Luna
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Yes, I have deduced this method, but I am just stuck on how should I explain it using strong Mathematical induction – Jake Blazer Apr 21 '20 at 01:29
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I mean technically you can use strong induction without using the "strong" part of the hypothesis. I don't see how much that would add; it proves unnecessary here. Any strong induction argument you make will be reducible to this simple induction. – Derek Luna Apr 21 '20 at 01:30