## Homework (22 Oct 2011)

Here’re the problems:

1. A square integer N ends in 4 identical digits d in its decimal representation, where $1 \le d \le 9$. Find all possible values of d. For each admissable value of d, find a possible N.
2. N is a perfect square whose second-to-last digit is odd. What can its last digit be?
3. Find all positive integers (x, y) for which $\frac 1 x - \frac 1 y = \frac 1 6$.
4. Solve for $y^4 = x^4 + 45x^2$, where x and y are positive integers.
5. Prove that the congruence $x^2 \equiv 1 \pmod {2^r}$, $r \ge 3$ has exactly 4 solutions modulo $2^r$.
That’s all. 🙂
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