We have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.
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.