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Number Theory

Number Theory Warmups

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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