The idea of Public Key Cryptography is to send messages in such a
  way that only the person who receives the messages can understand
  them. Even if everyone knows the method of encryption nobody
  except the receiver has all the information needed in order to
  decypher the message. To solve this problem you are in the
  position of Bob receiving a secret number from Alice who has sent
  him the coded message 180. Bob is the only person in the world
  who knows that he has to use 59 in order to find the secret
  number $M$ which Alice has sent to him, and which he knows to be
  between 0 and 390. <br />
  <br />
  Bob has to find the number equivalent to $180^{59}\pmod {391}$ so
  can you do this for him? <br />
  <br />
  <br />

  <div class="framework">
    <br />
    NOTES AND BACKGROUND<br />
    <br />
    So that you can decipher the secret number using only a
    calculator the example given involves only small numbers like
    59 and 391 whereas very big numbers are involved when the
    method is used in the real world. Instead of 391, the product
    of two very large prime numbers would be used in real world
    applications of this method. The reason that Bob, and nobody
    else, knows to use the number 59 to decode the message is that
    it is derived from one of the factors of 391 and in real world
    applications it is impossible to find the factors of the very
    large numbers that are used.<br />
    <br />
    The article '<a href=
    "http://nrich.maths.org/public/viewer.php?obj_id=2200" title=
    "Excellent article with suggested further reading">Public Key
    Cryptography</a> ' gives a detailed explanation of how the
    method works and gives you help in working with modulus
    arithmetic.<br />
    This wikipedia page might also be of interest <a href=
    "http://en.wikipedia.org/wiki/Cobalt-59">http://en.wikipedia.org/wiki/Cobalt-59</a>
  </div><br />
  <br />