Proving 8 Digit Number Divisible by 137 by Induction

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To prove that an 8-digit number of the form abcdabcd is divisible by 137, start by establishing the base case with the smallest 8-digit number, 10001, which is divisible by 137. Assume that for a four-digit number A, the number formed by repeating A is divisible by 137. The next step involves creating a formula for this repeated number and demonstrating that when 1 is added to A, the resulting number remains divisible by 137. This approach utilizes mathematical induction to validate the divisibility for all such 8-digit numbers. The discussion emphasizes the importance of clearly defining the induction hypothesis and the transition to the k+1 case.
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Homework Statement



Given a "8 digit number" abcdabcd.That means 1st digit =5th digit, 2nd digit =6th digit etc. Also, note that the smallest possible "8 digit number" is 10001(i.e.00010001). Prove this by induction that it is divisible by 137.

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The Attempt at a Solution



I can prove this actually but not able to find it by induction. I don't know how to present and start the assumption and k+1 phase.

Thanks for helping :)
 
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The smallest one is given, so you start by showing that 10001 is divisible by 137.
Suppose now that for some four digit number (possibly padded by zeros from the left) A, the assumption is true, that is the number obtained by writing A twice in a row is divisible by 137. Can you write a formula for this number? Then the induction step consists of adding 1 to A and showing that the difference is divisible by 137.
 

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