If one starts with the key of "C" as a central point and builds a major
scale from the 5th note (G), to have the "correct" sound of a major scale,
a sharp must be placed at the 7th degree. Likewise, for a scale starting
at the 4th note of C (F), a flat must be placed at the 4th degree. 
Notice how, just like in math, the new sharp or flat is carried over to
the next key in the cycle!

  new "b"at 4th note   C  D  E  F  G  A  B  C        new "#" at 7th note
                   |  ~~~~~~~~~~~|~~~~~~~~~~~~       |
 4THS     F  G  A  Bb C  D  E  F | G  A  B  C  D  E  F# G    5THS
  |       Bb C  D  Eb F  G  A  Bb| D  E  F# G  A  B  C# D     |
 \|/      Eb F  G  Ab Bb C  D  Eb| A  B  C# D  E  F# G# A    \|/
          Ab Bb C  Db Eb F  G  Ab| E  F# G# A  B  C# D# E
 /|\      Db Eb F  Gb Ab Bb C  Db| B  C# D# E  F# G# A# B    /|\
  |       Gb Ab Bb Cb Db Eb F  Gb| F# G# A# B  C# D# E# F#    |
 5THS     Cb Db EB Fb Gb Ab Bb Cb| C# D# E# F# G# A# B# C#   4THS

       NOTE: The 4ths and 5ths are opposites of each other-
         F is the 4th of C, and C is the 5th of F -etc...
     Also, note how the cycles loop around to meet each other
  For example- Gb and F# are the same pitches and, if you continue
   say, from F# to C#, that is the same as going from Gb to Db...

     Also, if you continue beyond C# to G#..., or Cb to Fb, you'll find
   yourself in the realm of "double" sharps(##) and flats(bb)!
       NOTE: "##" is usually written as "X"