Prove that the product of any three consecutive natural numbers

Thread starter 1MileCrash
Start date Oct 4, 2011
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Natural Natural numbers Numbers Product

In summary, consecutive numbers are numbers that follow one after the other in a sequence. To prove that the product of any three consecutive natural numbers is divisible by 6, we can use the formula n * (n+1) * (n+2) and see that one number will always be a multiple of 3 and one will always be a multiple of 2, making the product divisible by 6. This proof holds true for any set of three consecutive natural numbers and is significant in demonstrating mathematical patterns and rules, as well as the relationship between multiplication and division. It is also a fundamental concept in number theory.

Oct 4, 2011

1MileCrash

Homework Statement

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

Homework Equations

The Attempt at a Solution

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Oct 4, 2011

CompuChip

Science Advisor

Homework Helper

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

Oct 4, 2011

flyingpig

consecutive means like in order, i.e.

"1,2,3,4,5,6..."

1. What does it mean for three numbers to be consecutive?

Consecutive numbers are numbers that are in a sequence and follow one after the other. For example, 1, 2, 3 are consecutive numbers because they are in a sequence and follow each other.

2. How do you prove that the product of any three consecutive natural numbers is always divisible by 6?

To prove that the product of any three consecutive natural numbers is divisible by 6, we can use the formula for the product of consecutive natural numbers: n * (n+1) * (n+2). By substituting any natural number for n, we can see that the product will always be divisible by 6 because one of the numbers in the product will always be a multiple of 3 and one will always be a multiple of 2, making the product divisible by 6.

3. Can you provide an example to illustrate this proof?

Sure, let's take the consecutive natural numbers 4, 5, and 6. When we plug these numbers into our formula, we get: 4 * (4+1) * (4+2) = 4 * 5 * 6 = 120. Since 120 is divisible by both 3 and 2, we can see that the product of any three consecutive natural numbers is always divisible by 6.

4. Does this proof hold true for any set of three consecutive natural numbers?

Yes, this proof holds true for any set of three consecutive natural numbers because the formula n * (n+1) * (n+2) will always result in a product that is divisible by 6.

5. What is the significance of this proof in mathematics?

This proof is significant because it demonstrates a mathematical pattern and rule that can be applied to any set of three consecutive natural numbers. It also highlights the relationship between multiplication and division, as well as the properties of consecutive numbers. This proof can be used to solve various mathematical problems and is a fundamental concept in number theory.