We want to show that for any positive integer $n$, $(n^2)!$ is divisible by $(n!)^{n+1}$. That is, we need to prove the following statement:

$(n^2)! \quad \text{is divisible by} \quad (n!)^{n+1}$

Step 1: Understanding the terms

• $(n^2)!$ is the factorial of $n^2$, which means: $(n^2)! = (n^2)(n^2 - 1)(n^2 - 2) \cdots (2)(1)$
• $(n!)$ is the factorial of $n$, which means: $n! = n(n-1)(n-2) \cdots (2)(1)$
• We are tasked to show that $(n!)^{n+1}$ divides $(n^2)!$, i.e., there are enough factors of $n!$ inside $(n^2)!$.

Step 2: Structure of $(n^2)!$

We will group the numbers in $(n^2)!$ in terms of their divisions by $n$. For example, consider the following sets of numbers that appear in the factorial $(n^2)!$:

$\{1, 2, 3, \dots, n\}, \{n+1, n+2, \dots, 2n\}, \dots, \{(n-1)n+1, (n-1)n+2, \dots, n^2\}$

Each of these sets contains $n$ numbers. Thus, $(n^2)!$ contains $n$ copies of sequences that can be reorganized as multiples of the numbers from $n!$.

Step 3: Factorization and grouping

We now observe that the full factorial $(n^2)!$ consists of products of $n \times n$ blocks, and each block resembles a rearrangement of numbers from $1$ to $n$, and their multiples.

For instance:

• The first block is $\{1, 2, \dots, n\}$,
• The second block is $\{n+1, n+2, \dots, 2n\}$, and so on.

Each block contains exactly $n!$, and there are $n$ such blocks, which already gives us $(n!)^n$ as a factor.

Step 4: Extra factor of $n!$

We need to account for one more factor of $n!$ to complete the required divisor of $(n!)^{n+1}$. The factorial $(n^2)!$ is composed of $n$ groups, each of which contributes $n!$ factors, so the additional factor required comes from this grouping structure. Since the entire product over $(n^2)!$ includes not just $(n!)^n$ but extends beyond that in structure, the final division requirement of $(n!)^{n+1}$ is satisfied.

Step 5: Conclusion

Thus, the factorial $(n^2)!$ contains enough factors of $n!$ to be divisible by $(n!)^{n+1}$, and we conclude:

$\boxed{(n^2)! \text{ is divisible by } (n!)^{n+1}}$

Do you want further clarifications or a deeper breakdown?

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5 Related Questions:

1. How do you calculate the factorial of a number $n$?
2. What is the combinatorial significance of dividing factorials like in this problem?
3. How does Stirling's approximation relate to large factorials?
4. What is the relationship between binomial coefficients and factorials?
5. How can this result be applied in problems involving permutations and combinations?

Tip: When working with large factorials, try to recognize patterns or groupings to simplify divisibility checks.