Are all three digit numbers divisible by eleven and other questions

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I have explained some short cuts to determine when a number is divisible by three, by seven, by nine, and by eleven. The magic of google brings me visitors that have asked questions with the same words, but not the same meaning.

For example, one searcher wonders if all three digit numbers are divisible by eleven. The answer is no. But here is a proof.

Let’s use a method called proof by contradiction. If you assume something is true, and that assumption implies something is true, but you can tell that it is false, then you can conclude that your original assumption is false.

So, let us assume that all three digit numbers are divisible by eleven. What does that imply? I can add one to most three digit numbers, and still have a three digit number. 999 is the only three digit numbers for which adding one does not give us another three digit number. If I have a number that is divisible by eleven, and add one to it, I have a number that has a remainder of one when divided by eleven. Since a number that has a remainder of one when divided by eleven is not divisible by eleven, and our assumption that all three digit numbers are divisible by eleven implied that adding one to all three digit numbers under 999  would still be divisible by eleven, we have a contradiction. Therefore, we have proven that not all three digit numbers are divisible by eleven.

How about another proof. The statement makes a claim of all of something. So if we can find just one counter example, we can disprove the over reaching statement. So how about one hundred. 100 divided by 11 is 9 and 1/11th. That is not an even number, so 100 is a counter example that disproves the claim.

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