PUMAC 2008-2009 Number Theory A problems

1. (2 points) How many zeros are there at the end of 792! when written in base 10?

2. (3 points) Find all integral solutions to $x^y-y^x=1$ .

3. (3 points) Find the largest integer $n$ , where $2009^n$ divides $2008^{2009^{2010}}+2010^{2009^{2008}}$ .

4. (3 points) $f(n)$ is the sum of all integers less than $n$ and relatively prime to $n$ . Find all integers $n$ such that there exist integers $k$ and $l$ such that $f(n^k)=n^l$ .

5. (4 points) If $f(x)=x^{x^{x^{x}}}$ , find the last two digits of $f(17)+f(18)+f(19)+f(20)$ .

6. (4 points) What is the largest integer which cannot be expressed as $2008x+2009y+2010z$ for some positive integers $x$ , $y$ , and $z$ ?

7. (5 points) Find the smallest positive integer $n$ such that $32^n=167x+2$ for some integer $x$ .

8. (5 points) Find all sets of three primes $p$ , $q$ , and $r$ such that $p+q=r$ and $(r-p)(q-p)-27p$ is a perfect square.

9. (7 points) Find the number of positive integer solutions of $(x^2+2)(y^2+3)(z^2+4)=60xyz$ .

10. (7 points) What is the smallest number $n$ such that you can choose $n$ distinct odd integers $a_1, a_2,...a_n$ , none of them 1, with $\dfrac{1}{a_1}+\dfrac{1}{a_2}+...+\dfrac{1}{a_n}=1$ ?

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PUMAC 2008-2009 Number Theory A problems