Number Theory
# Number Theory Warmups

\[\large \frac{1}{a}+\frac{1}{b}=\frac{1}{100000}\]

How many distinct ordered pairs of positive integers \((a,b)\) are there which satisfy the above equation?

\[\large \displaystyle \dfrac{a}{\frac{b}{c}} = \dfrac{\frac{a}{b}}{c}\]

The above equation is a common mistake made when interpreting fractions.

How many ordered triplets of integers \({(a,b,c)}\) with \(-10\leq a,b,c \leq 10\) are there, such that the above equation is a true statement?

Given a positive integer \(n\), let \(p(n)\) be the product of the non-zero digits of \(n\). (If \(n\) has one digit, then \(p(n)\) is equal to that digit.) Let

\[S = p(1) + p(2) + \cdots + p(999). \]

What is the largest prime factor of \(S\)?

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