### Purr-fection

What is the smallest perfect square that ends with the four digits 9009?

### Old Nuts

In turn 4 people throw away three nuts from a pile and hide a quarter of the remainder finally leaving a multiple of 4 nuts. How many nuts were at the start?

### Mod 7

Find the remainder when 3^{2001} is divided by 7.

# Double Time

##### Stage: 5 Challenge Level:

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.

 a b c d e f g h i j k l m n o p q r s t u v w x y z 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

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 $(\alpha,\beta)$

 $C$ $\alpha'$ $\beta'$ $\alpha$ $\beta$ $P$ dj 3 9 8 7 ih lb 11 1 0 21 av rn 17 13 4 13 en qm 16 12 14 18 os bu 1 20 15 4 pe ao 0 14 2 8 gi hd 7 3 0 11 al eo 4 14 6 8 gi kr 10 17 5 19 ft ia 8 0 8 0 ia cs 2 18 12 14 mo ud 20 3 13 11 nl rx 17 23 24 15 yp cm 2 12 0 18 as qo 16 14 18 8 si bn 1 13 14 13 on fr 5 17 0 19 at ld 11 3 4 11 el ek 4 10 24 2 yc th 19 7 20 17 ur ys 24 18 8 14 io wm 22 12 20 18 us

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'.