Prove that the product of any three consecutive natural numbers

Click For Summary
SUMMARY

The product of any three consecutive natural numbers is always divisible by 6. This is established through mathematical induction, which demonstrates that for any integer n, the expression n(n+1)(n+2) results in a product that includes at least one even number and one multiple of 3. Therefore, the product is guaranteed to be divisible by both 2 and 3, confirming divisibility by 6.

PREREQUISITES
  • Understanding of mathematical induction
  • Basic knowledge of natural numbers
  • Familiarity with divisibility rules
  • Ability to manipulate algebraic expressions
NEXT STEPS
  • Study the principles of mathematical induction in detail
  • Explore the properties of natural numbers and their sequences
  • Learn about divisibility rules, particularly for 2 and 3
  • Practice proving statements using induction with various examples
USEFUL FOR

Students studying mathematics, particularly those focusing on number theory and proof techniques, as well as educators looking to reinforce concepts of divisibility and induction.

1MileCrash
Messages
1,338
Reaction score
41

Homework Statement



Prove that the product of any three consecutive natural numbers is divisible by 6.

Homework Equations





The Attempt at a Solution




wat
 
Physics news on Phys.org


You said "induction" in your title.
So.. did you try using induction?
 


consecutive means like in order, i.e.

"1,2,3,4,5,6..."
 

Similar threads

Can Natural Numbers m and n Satisfy x + 1/m < y - 1/n?
  • · Replies 22 ·
Replies
22
Views
2K
Consecutive odd natural numbers - one is composite. Prove
  • · Replies 1 ·
Replies
1
Views
4K
Prove that the product of 4 consecutive numbers cannot be a perfect square
  • · Replies 14 ·
Replies
14
Views
5K
Expected number of coin flips before N consecutive heads?
  • · Replies 15 ·
Replies
15
Views
7K
Intervals with Natural Number Endpoints
  • · Replies 3 ·
Replies
3
Views
1K
Prove that ##\lim\limits_{x \to \infty} f(x) = 0##
  • · Replies 3 ·
Replies
3
Views
2K
Prove that the product of 2 consecutive natural numbers is even
  • · Replies 2 ·
Replies
2
Views
5K
High School Cool fact about number of digits in n!
  • · Replies 3 ·
Replies
3
Views
2K
Prove that the product of any three consecutive integers is
  • · Replies 11 ·
Replies
11
Views
4K
Can the Product of n Consecutive Positive Integers Be Divisible by n!?
  • · Replies 4 ·
Replies
4
Views
3K