We are tasked with finding the greatest possible value of $n$ such that when the product of $n$ consecutive integers is divided by 7, the remainder is a positive number.

Problem Restatement:

• We have a sequence of $n$ consecutive integers, and their product is denoted as: $P = x(x+1)(x+2) \dots (x+n-1)$
• When this product is divided by 7, the remainder must be a positive number (i.e., not 0), implying that the product is not divisible by 7.
• We are tasked with finding the largest $n$ such that $P$ is not divisible by 7.

Step-by-Step Solution:

1. Understanding Modulo 7:

If $P$ is divisible by 7, then at least one integer in the sequence $x, x+1, x+2, \dots, x+n-1$ must be divisible by 7. To avoid divisibility by 7, none of these integers can be divisible by 7.

2. Considering the Range:

For $P$ to not be divisible by 7, none of the integers in the sequence can be a multiple of 7. However, since we are dealing with $n$ consecutive integers, if $n \geq 7$, one of the integers in the sequence will always be divisible by 7 because of the properties of consecutive integers (every set of 7 consecutive integers contains exactly one multiple of 7).

3. Maximizing $n$:

The largest $n$ such that no integer in the sequence is divisible by 7 is $n = 6$. In any sequence of 6 consecutive integers, there may not be a multiple of 7, but once $n$ reaches 7, one of the integers must be divisible by 7.

4. Conclusion:

The greatest possible value of $n$ such that the product of $n$ consecutive integers is not divisible by 7 is $\boxed{6}$.

Would you like more details or have any questions? Here are 5 related questions to expand on this:

1. What is the general rule for the divisibility of a product of consecutive integers by a prime number?
2. How does the inclusion of zero in a sequence affect the divisibility of the product?
3. What is the remainder when a product of integers is divided by a composite number like 12?
4. Can you generalize this approach for any prime number $p$?
5. How would this problem change if we were considering $k = 10$ instead of 7?

Tip: When working with consecutive integers and divisibility, consider the number's prime factors and the intervals between them to identify divisibility patterns.