# Number Theory

 1. (3 p.) $$n$$ is an integer between 100 and 999 inclusive, and $$n^{\prime}$$ is the integer formed by reversing the digits of $$n$$. How many possible values are for $$|n-n^{\prime}|$$?

 2. (38 p.) Find the largest possible integer $$n$$ such that $$\sqrt n + \sqrt{n+60} = \sqrt m$$ for some non-square integer $$m$$.

 3. (30 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 $$a + b + c + d$$.

 4. (7 p.) How many pairs of integers $$(x,y)$$ are there such that $$x^2-y^2=2400^2$$?

 5. (19 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 $$6^n$$.

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