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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.

##### Age 16 to 18 Challenge Level:

Here is another brilliant solution from Andrei from School No. 205, Bucharest, Romania

I have to find the secret message

20 14 19 23 11 13 20 21 4 5 11 23 18 6 19 14 19 4 13 21 24 16 19 20 14 21 4 7 17 24 11 1 20 20 14 19 15 11 6 16 12 21 13 20 14 17 20 21 20 21 13 5 11 23 18 6 19 14 19 4 13 21 24 16 19

which has been coded by using the equation: $$C=7P + 17 \pmod {26}$$ where $C$ is the cipher value of the corresponding $P$ and $P$ represents the letters of the alphabet taking $a=0,\ b=1,\ c=2,\ ...$ to $z=25$.

Solving this equation for $P$ gives $$P={1\over 7}C - {17\over 7} \pmod{26}.$$ Using the reverse of the multiplication table mod 26, I found: $$P = 15C - 21 = 15 C + 5 \pmod{26} \quad (1)$$ Performing the operations in equation (1) using the properties of arithmetic operations mod 26, I found the following sequence of numbers 19, 7, 4, 12, 14, 18, 19, 8, 13, 2, 14, 12, 15, 17, 4, 7, 4, 13, 18, 8, 1, 11, 4, 19, 7, 8, 13, 6, 26, 1, 14, 20, 19,19, 7, 4, 22, 14, 17, 11, 3, 8, 18, 19, 7, 26, 19, 26, 18, 2, 14, 12, 15, 17, 4, 7, 4, 13, 18, 8, 1, 11, 4. These, transformed into words, produced the following message:

"The most incomprehensible thing about the world is that it is comprehensible".