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Number Theory Warmups

If numbers aren't beautiful, we don't know what is. Dive into this fun collection to play with numbers like never before, and start unlocking the connections that are the foundation of Number Theory.

Number Theory Warmups: Level 4 Challenges

         

What is the remainder when \[1^{2013}+2^{2013}+\cdots +2012^{2013}+2013^{2013}\] is divided by \(2014\)?

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

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 \(S\)?

\( a, b\) and \(c\) are distinct positive integers strictly greater than 1. If \( abc \) divides \( (ab-1)(bc-1)(ca-1) \), what is the value of \(abc\)?

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