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Number Theory Warmups
What is the remainder when \[1^{2013}+2^{2013}+\cdots +2012^{2013}+2013^{2013}\] is divided by \(2014\)?
by
Daniel Liu
\[\large \frac{1}{a}+\frac{1}{b}=\frac{1}{100000}\]
How many distinct ordered pairs of positive integers \((a,b)\) are there which satisfy the above equation?
by
Yash Singhal
\[\large \displaystyle \dfrac{a}{\frac{b}{c}} = \dfrac{\frac{a}{b}}{c}\]
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?
by
Aditya Raut
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 \(S\)?
by
Sharky Kesa
\( a, b\) and \(c\) are distinct positive integers strictly greater than 1. If \( abc \) divides \( (ab-1)(bc-1)(ca-1) \), what is the value of \(abc\)?
by
Calvin Lin