To solve this problem, we need to determine the largest integer $n$ for which the product

$(b - a)(c - a)(d - a)(c - b)(d - b)(d - c)$

is guaranteed to be a multiple. Given the hint, let's set $x = b - a$, $y = c - b$, and $z = d - c$. Then the product can be rewritten as:

$xyz(x + y)(x + y + z)(y + z)$

Step 1: Understanding the factors

Each term represents a positive difference between two of the integers $a$, $b$, $c$, and $d$:

• $x = b - a$
• $y = c - b$
• $z = d - c$
• $x + y = c - a$
• $x + y + z = d - a$
• $y + z = d - b$

Step 2: Analyze prime factorizations

We need to consider the prime factorizations of these terms. Notice that the product $xyz(x + y)(x + y + z)(y + z)$ will always include these terms:

• $x$, $y$, and $z$ are differences between consecutive integers, so each can be any positive integer.
• The terms $x + y$, $x + y + z$, and $y + z$ are sums of these differences, so they also vary with the specific values of $a$, $b$, $c$, and $d$.

Step 3: Evaluate modulo prime numbers

We should check the divisibility of this product by small prime numbers, beginning with 2, 3, 4, 6, and 12.

Divisibility by 2:

At least one of $x$, $y$, or $z$ must be even, so the product is always divisible by 2.

Divisibility by 3:

By the Pigeonhole Principle (mod 3), at least one of $x$, $y$, $z$, $x+y$, $y+z$, or $x+y+z$ must be divisible by 3.

Divisibility by 4:

We consider divisibility by 4. If $x$, $y$, or $z$ is even, then at least one of $x$, $y$, $z$, or $x+y$, $x+y+z$, $y+z$ is divisible by 4, ensuring the product is divisible by 4.

Divisibility by 6:

The product is divisible by 2 and by 3, so it is divisible by 6.

Divisibility by 12:

To ensure divisibility by 12, we must check if the product is divisible by both 4 and 3, which is already true from the above analysis.

Step 4: Conclusion

The product $(b - a)(c - a)(d - a)(c - b)(d - b)(d - c) = xyz(x + y)(x + y + z)(y + z)$ is always divisible by 12, no matter the positive integers $a < b < c < d$.

Thus, the largest integer $n$ for which this product is guaranteed to be a multiple is:

$\boxed{12}$

Would you like more details on any specific step, or do you have any questions?

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Related Questions:

1. What is the significance of the Pigeonhole Principle in modular arithmetic?
2. How does the choice of prime factors affect the divisibility of a product?
3. Why is considering consecutive differences an effective strategy in problems like this?
4. Can you generalize this result to more than four integers?
5. How would the result change if $a, b, c, d$ were not required to be positive integers?
6. What role does the modulus play in finding the largest guaranteed divisor?
7. How can this problem be adapted to different arithmetic sequences?
8. What are some other mathematical problems that involve finding the greatest common divisor of products?

Tip: When determining the divisibility of a product, consider the smallest prime factors and combinations of factors that consistently appear across all possible cases. This can often lead to identifying the largest guaranteed divisor.