# Number Theory

 1. (4 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 $$n$$.

 2. (47 p.) Find the largest possible integer $$n$$ such that $$\sqrt n + \sqrt{n+60} = \sqrt m$$ for some non-square integer $$m$$.

 3. (23 p.) Find the least positive integer $$n$$ such that when its leftmost digit is deleted, the resulting integer is equal to $$n/29$$.

 4. (14 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 $$a + b + c$$.

 5. (9 p.) How many pairs of integers $$(x,y)$$ are there such that $$x^2-y^2=2400^2$$?

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