Proof of Divisibility of 8 rule

  • Thread starter Thread starter smiles988
  • Start date Start date
  • Tags Tags
    Divisibility Proof
Click For Summary
SUMMARY

The discussion centers on proving that if a natural number \( n \) has its last three digits divisible by 8, then \( n \) itself is divisible by 8. The proof approach suggested is a direct proof, starting from the assumption that the last three digits of \( n \) are divisible by 8. The key insight is recognizing that 1000, the lowest power of 10 divisible by 8, plays a crucial role in the proof structure.

PREREQUISITES
  • Understanding of natural numbers and their properties
  • Basic knowledge of divisibility rules, specifically for 8
  • Familiarity with direct proof techniques in mathematics
  • Concept of powers of 10 and their divisibility
NEXT STEPS
  • Study the properties of divisibility, focusing on divisibility by 8
  • Learn about direct proof methods in mathematical logic
  • Explore the significance of powers of 10 in number theory
  • Investigate other divisibility rules for different integers
USEFUL FOR

Students studying number theory, mathematics educators, and anyone interested in understanding divisibility rules and proof techniques.

smiles988
Messages
6
Reaction score
0

Homework Statement


Let n be a natural number. If the number formed by the last three digits of n is divisible by 8, then n is divisible by 8.


Homework Equations


Natural numbers are the set of {1,2,3,4,5,6,...}


The Attempt at a Solution


I believe we should use a direct proof to prove this conditional statement. Thus we will assume that n is a natural number and the number formed by the last three digits of n is divisible by 8. We will prove that n is divisible by 8.
I don't know where to go from here. Please help!
 
Physics news on Phys.org
Hint: What is the lowest power of 10 that is divisible by 8?
 
Reply to Hint

Okay so 1000 is the lowest power of 10 that is divisible by 8. How do I use that?
 

Similar threads

Base 8 Logic Puzzle: Finding Solutions to Divisibility Rules
  • · Replies 9 ·
Replies
9
Views
2K
Show That If Alt Sum of Digits Div By 11, n Is Divisible By 11
  • · Replies 1 ·
Replies
1
Views
3K
Prove True/False that n^3-n is Always Divisible By 6
  • · Replies 4 ·
Replies
4
Views
3K
Discrete Math Proof: Necessary Condition for Divisibility by 6
  • · Replies 3 ·
Replies
3
Views
4K
The only prime of the form n^3-1 is 7?
  • · Replies 2 ·
Replies
2
Views
4K
Prove every subset of countable set is either finite or else countable
  • · Replies 1 ·
Replies
1
Views
2K
Proof: Divisibility of Integers by 4
  • · Replies 5 ·
Replies
5
Views
2K
Proof That an Integer is Divisible by 2
  • · Replies 3 ·
Replies
3
Views
2K
Root test proof using Law of Algebra
  • · Replies 6 ·
Replies
6
Views
2K
Proving n^2 is Divisible by 4: Contradiction Method
  • · Replies 13 ·
Replies
13
Views
4K