Proving 8 Digit Number Divisible by 137 by Induction

  • Thread starter Thread starter haoku
  • Start date Start date
  • Tags Tags
    Induction
Click For Summary
SUMMARY

The discussion focuses on proving that an 8-digit number of the form abcdabcd is divisible by 137 using mathematical induction. The smallest 8-digit number, 10001, serves as the base case for the proof. The participants suggest starting with the assumption that a four-digit number A, when repeated, is divisible by 137, and then progressing to show that the next number, A+1, maintains this divisibility. The key to the proof lies in formulating the number and demonstrating the induction step effectively.

PREREQUISITES
  • Understanding of mathematical induction
  • Familiarity with divisibility rules, specifically for 137
  • Basic algebraic manipulation and number theory
  • Knowledge of constructing sequences and patterns in numbers
NEXT STEPS
  • Study mathematical induction techniques in detail
  • Explore divisibility rules for numbers, focusing on 137
  • Learn how to formulate sequences and their properties
  • Practice problems involving number patterns and proofs
USEFUL FOR

Students studying number theory, mathematics educators, and anyone interested in mathematical proofs and induction techniques.

haoku
Messages
24
Reaction score
0

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.

Homework Equations





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 :)
 
Physics news on Phys.org
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.
 

Similar threads

What is the Sixth Digit of a Number That is a Multiple of 73 and 137?
  • · Replies 4 ·
Replies
4
Views
2K
How Many 5-Digit Numbers from 4, 5, 6, 8, 9 are Divisible by 8?
  • · Replies 4 ·
Replies
4
Views
2K
Product of Missing Digits in a Number
  • · Replies 2 ·
Replies
2
Views
2K
Proof via mathematical induction
  • · Replies 17 ·
Replies
17
Views
2K
Prove that there are 3^n n-digit numbers formed with 4, 5 ,6
  • · Replies 4 ·
Replies
4
Views
2K
A simple combinatorics problem
  • · Replies 7 ·
Replies
7
Views
2K
Solving Odd Digit Divisibility by Five using Permutations
  • · Replies 8 ·
Replies
8
Views
2K
Prove ##5^n+9<6^n## for ##n\epsilon N|n\ge2## by induction
  • · Replies 5 ·
Replies
5
Views
3K
MHB How Many Numbers Less Than 1000 Divisible by 5 Are Formed Using Unique Digits?
  • · Replies 1 ·
Replies
1
Views
2K
Prove Induction: u_n < 4 for All n ≥ 1
  • · Replies 24 ·
Replies
24
Views
3K