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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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