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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:

Mathematical methods of encryption are of vital importance in ensuring the security of electronic communication and financial transactions.

This is an example of a simple cipher which can be cracked quite easily to serve as an introduction to some of the ideas.

The message given has been enciphered using the formula $C=7P+17 \pmod { 26}$ where $P$ represents the letters of the alphabet taking values $a=0,\ b=1,\ {\rm to}\ z=25$ and $C$ represents the cipher value of the corresponding $P$.

It is easy to decipher the message by using the given formula to find the cipher numbers for each letter. But can you rearrange the formula to give $P$ in terms of $C$ using the multiplicative inverse of 7 (mod 26) and the additive inverse of 17 (mod 26) and hence decipher the 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

You might like to write a computer program to encipher or decipher messages using this system.