Question 13

The product of any three consecutive natural numbers is divisible by 6 (True/false).

Solution:

To determine whether the product of any three consecutive natural numbers is divisible by 6, we can follow these steps:

Step 1: Define the three consecutive natural numbers
Let the three consecutive natural numbers be represented as:
– $$n$$
– $$n+1$$
– $$n+2$$

Step 2: Calculate the product of these three numbers
The product of these three consecutive numbers can be expressed as:
$$P=n\times (n+1)\times (n+2)$$

Step 3: Check divisibility by 2
Among any three consecutive natural numbers, at least one of them must be even. This is because consecutive numbers alternate between odd and even. Therefore, the product $$P$$ will always include at least one factor of 2.

Step 4: Check divisibility by 3
Among any three consecutive natural numbers, at least one of them must be divisible by 3. This is due to the fact that every third natural number is a multiple of 3. Therefore, the product $$P$$ will also include at least one factor of 3.

Step 5: Conclusion about divisibility by 6
Since the product $$P$$ contains at least one factor of 2 and one factor of 3, it follows that:
$$P\text{is divisible by}2\times 3=6$$

Thus, we conclude that the product of any three consecutive natural numbers is indeed divisible by 6.

Final Answer
True: The product of any three consecutive natural numbers is divisible by 6.