Prove That the Product of Three Consecutive Positive Integers Is Divisible by 6

Video Explanation

Question

Prove that the product of three consecutive positive integers is divisible by 6.

Solution

Step 1: Let the Three Integers Be

Let the first positive integer be \(n\), where \(n\) is a positive integer. Then the next two consecutive integers are:

\[ n,\quad n+1,\quad n+2 \]

Step 2: Consider the Product

The product of these three integers is:

\[ n(n+1)(n+2) \]

Step 3: Show Divisibility by 2

Among any three consecutive integers, at least one must be even. Therefore, the product contains an even factor, which means the product is divisible by 2.

Step 4: Show Divisibility by 3

Now consider divisibility by 3. Among any three consecutive integers, one of them must be divisible by 3 (because every third integer is divisible by 3). Therefore, the product has a factor divisible by 3.

Step 5: Combine the Results

Since the product is divisible by both 2 and 3, it must be divisible by their least common multiple, which is 6.

Conclusion

Hence, the product of three consecutive positive integers is divisible by 6.

\[ \therefore \quad \text{Proved.} \]