Double Time

This solution comes from Andrei from School No. 205, Bucharest, Romania.

To solve this problem I followed the following steps:

- associating to the letters of the alphabet numbers between 0 and 25, I transformed the coded message into a set of pairs of numbers $(\alpha',\beta')$

- I solved the system of equations for $(\alpha,\beta)$ in terms of $(\alpha',\beta')$.

- I used the same association as in the first step, and I transformed the set of numbers $(\alpha, \beta)$ into letters, and I found the message.

For each pair of numbers $(\alpha', \beta')$ I have to solve the system to determine $(\alpha, \beta)$ $$ \alpha'= \alpha + 3\beta\pmod {26}\quad (1) $$ $$\beta' = 5\beta \quad \pmod{26}\quad (2)$$

I start from the last equation: $$\beta = {1\over 5}\beta' \pmod{26}$$ To determine 1/5 (mod 26), I first constructed the table of multiplication for 5 (mod 26) to see where I obtain 1. As 21 multiplied by 5 gives 1 (mod 26) it follows that 1/5 (mod 26) is 21. This means: $$\beta = 21\beta' \pmod{26}\quad (3)$$ and for $\alpha$ I obtained successively $$\alpha = \alpha'-3\beta = \alpha'-3\times 21\beta' = \alpha' - 11\beta' \pmod{26} $$ that is $$\alpha = \alpha' +15\beta' \pmod{26} \quad (4).$$ Now, the sequence of numbers $(\alpha', \beta')$ is transformed by equations (3) and (4) into the sequence $(\alpha,\beta)$

The message could be read as the quotation from Einstein talking about himself as a mathematician:'I have no special gift. I am only passionately curious'.