# Number Theory

 1. (17 p.) Find the least positive integer $$n$$ such that when its leftmost digit is deleted, the resulting integer is equal to $$n/29$$.

 2. (10 p.) Let $$a$$, $$b$$, $$c$$ be positive integers forming an increasing geometric sequence such that $$b-a$$ is a square. If $$\log_6a + \log_6b + \log_6c = 6$$, find $$a + b + c$$.

 3. (3 p.) Let $$n$$ be the largest positive integer for which there exists a positive integer $$k$$ such that $k\cdot n! = \frac{(((3!)!)!}{3!}.$ Determine $$n$$.

 4. (27 p.) Let $$0 < a < b < c < d$$ be integers such that $$a$$, $$b$$, $$c$$ is an arithmetic progression, $$b$$, $$c$$, $$d$$ is a geometric progression, and $$d - a = 30$$. Find $$a + b + c + d$$.

 5. (41 p.) Let $$\tau (n)$$ denote the number of positive divisors of $$n$$, including 1 and $$n$$. Define $$S(n)$$ by $$S(n)=\tau(1)+ \tau(2) + \dots + \tau(n)$$. Let $$a$$ denote the number of positive integers $$n \leq 2008$$ with $$S(n)$$ odd, and let $$b$$ denote the number of positive integers $$n \leq 2008$$ with $$S(n)$$ even. Find $$|a-b|$$.

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