Number Theory

Number Theory Warmups

Number Theory Warmups: Level 4 Challenges


What is the remainder when 12013+22013++20122013+201320131^{2013}+2^{2013}+\cdots +2012^{2013}+2013^{2013} is divided by 20142014?

1a+1b=1100000\large \frac{1}{a}+\frac{1}{b}=\frac{1}{100000}

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

abc=abc\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){(a,b,c)} with 10a,b,c10-10\leq a,b,c \leq 10 are there, such that the above equation is a true statement?

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

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

What is the largest prime factor of SS?

a,b a, b and cc are distinct positive integers strictly greater than 1. If abc abc divides (ab1)(bc1)(ca1) (ab-1)(bc-1)(ca-1) , what is the value of abcabc?


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