The expression given is:

$\lfloor n \cdot p \rfloor - 10\lfloor n \cdot p \rfloor$

Breaking it down:

1. $\lfloor n \cdot p \rfloor$ represents the floor function of $n \cdot p$, which means it is the greatest integer less than or equal to $n \cdot p$.

2. The second term is 10 times the same floor value, $10\lfloor n \cdot p \rfloor$.

Thus, the whole expression simplifies to:

$\lfloor n \cdot p \rfloor - 10\lfloor n \cdot p \rfloor = -9\lfloor n \cdot p \rfloor$

Conclusion:

The expression simplifies to:

$-9\lfloor n \cdot p \rfloor$

Would you like more details on how the floor function works or on a specific part of this solution?

Here are 5 related questions to explore:

1. What is the floor function and how does it differ from the ceiling function?
2. How does multiplying within a floor function affect the output?
3. Can you generalize this expression for other integer multipliers?
4. How does this expression behave for non-integer values of $n$ and $p$?
5. What happens if we apply a different mathematical function instead of the floor function?

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