Number Theory
# General Diophantine Equations

Note that \(7!=10 \times9 \times8\times7.\)

What is the largest possible value of \(n\) for which \(n!\) can be represented as a product of \(n-3\) consecutive integers?

How many ordered pairs of positive integers \( (A,B) \), each of which are between 1 and 100 inclusive, are there such that

\[ A^B = B^A ?\]

I choose two different integers, both greater than 1, whose sum is less than 100. I whisper this sum to Sam, and whisper their product to Paula. They then have the following conversation:

Paula: I don’t know the numbers.

Sam: I knew you didn’t. I don’t either.

Paula: Ah, now I know them.

Sam: Now I know them too.

What is the LARGER number that I chose?

\[ \left( 1+\dfrac1a\right)\left( 1+\dfrac1b\right)\left( 1+\dfrac1c\right)\left( 1+\dfrac1d\right)\left( 1+\dfrac1e\right) = 14.4 \]

Find the number of 5-tuples of positive integers \((a, b, c, d, e)\) that satisfy the equation above.

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