# Number Theory

 1. (36 p.) Let $$f$$ be a function defined along the rational numbers such that $$f(\tfrac mn)=\tfrac1n$$ for all relatively prime positive integers $$m$$ and $$n$$. The product of all rational numbers $$0< x< 1$$ such that $f\left(\dfrac{x-f(x)}{1-f(x)}\right)=f(x)+\dfrac9{52}$ can be written in the form $$\tfrac pq$$ for positive relatively prime integers $$p$$ and $$q$$. Find $$p+q$$.

 2. (7 p.) Find the sum of all positive integers of the form $$n = 2^a3^b$$ $$(a, b \geq 0)$$ such that $$n^6$$ does not divide $$6^n$$.

 3. (36 p.) Suppose $$m$$ and $$n$$ are positive integers with $$m> 1$$ such that the domain of the function $$f(x) = \text{arcsin}(\log_{m}(nx))$$ is a closed interval of length $$\frac{1}{2013}$$. Let $$S$$ be the smallest possible value of $$m+n$$. Find the remainder when $$S$$ is divided by $$1000$$.

 4. (11 p.) If the corresponding terms of two arithmetic progressions are multiplied we get the sequence 1440, 1716, 1848, ... . Find the eighth term of this sequence.

 5. (8 p.) Determine the number of positive integers with exactly three proper divisors each of which is less than 50. (1 is a proper divisor of every integer greater than 1)

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