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The key point is that the number of positive integral solutions of the given equation is equal to the number of factors of (n!)^2.
(Hint: rearrage the equation. You may need to check out Diophantine Equation)
Another important point is to compute the number of factors of (n!)^2 with a good time complexity. It turns out that you could build each factor from the more fundamental element: the prime factors of (n!)^2.
(Hint: determine all the prime factors of (n!)^2, which is all the prime numbers not greater than n, and their number of appearances)
Once you have all the prime factors of (n!)^2 and their number of appearances, the last step is just a simple Combinatorial Counting Problem.
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The key point is that the number of positive integral solutions of the given equation is equal to the number of factors of (n!)^2. (Hint: rearrage the equation. You may need to check out Diophantine Equation)
Another important point is to compute the number of factors of (n!)^2 with a good time complexity. It turns out that you could build each factor from the more fundamental element: the prime factors of (n!)^2. (Hint: determine all the prime factors of (n!)^2, which is all the prime numbers not greater than n, and their number of appearances)
Once you have all the prime factors of (n!)^2 and their number of appearances, the last step is just a simple Combinatorial Counting Problem.