Number Theory

 1. (18 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.

 2. (6 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. (27 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 $$|a-b|$$.

 4. (34 p.) It is given that $$181^2$$ can be written as the difference of the cubes of two consecutive positive integers. Find the sum of these two integers.

 5. (11 p.) Find the least positive integer $$n$$ such that when its leftmost digit is deleted, the resulting integer is equal to $$n/29$$.

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