Homework (22 Oct 2011)

Posted on October 27, 2011 by limsup

Here’re the problems:

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.
N is a perfect square whose second-to-last digit is odd. What can its last digit be?
Find all positive integers (x, y) for which $\frac 1 x - \frac 1 y = \frac 1 6$ .
Solve for $y^4 = x^4 + 45x^2$ , where x and y are positive integers.
Prove that the congruence $x^2 \equiv 1 \pmod {2^r}$ , $r \ge 3$ has exactly 4 solutions modulo $2^r$ .