
Problem 9a. 1 Given the equation $\frac{1}{|x-2|}=\frac{1}{|x-52 a|}$, where $a$ is a parameter.

a) Solve the equation.

b) If $a$ is the square of a prime number prove that the equation has a solution which is a compose integer.


Solution: After squaring and simple calculations we obtain (26a1) $x=(26 a-1)(26 a+1)$.

a) If $a=\frac{1}{26}$ then every $x \neq 2$ is a solution. If $a \neq \frac{1}{26}$ then the only solution is $x=26 a+1$.

b) Let $a=p^{2}$ where $p$ is a prime number. If $p=3$ then $x=235$ is not a prime. If $p \neq 3$ then $p=3 k \pm 1$ and $x=26(3 k \pm 1)^{2}+1=$ $3 A+27$, which is divisible by 3 .
