When you divide the product of n consecutive integers by a natural number K, the remainder is a positive number. Find the greatest possible value of n, if k=7

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.

When you divide the product of n consecutive integers by a natural number K, the remainder is a positive number. Find the greatest possible value of n, if k = 7.

Find the largest value of n such that the product of n consecutive integers, when divided by 7, leaves a positive remainder. This problem explores number theory, divisibility, and modular arithmetic, making it ideal for students in Grades 9-12.

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