The cycle of 5ths

The pattern of the cycle of 5ths is key (pun intended) to music theory

C major has no #'s or b's, but G - the 5th note of C major - has a sharp (F#). D - the 5th note of G adds a sharp (F# and C#) & so on

Sharps always come in the same order:- C > G (F#) > D (F#C#) > A (F#C#G#)> E (F#C#G#D#)

So each time we go up a 5th, the new key adds a sharp (#)

F major is a 5th down (or 4th up) from C: and has a flat (b):- 

F (Bb) > Bb (Bb, Eb) > Eb (Bb,Eb,Ab) > Ab ( Bb, Eb, Ab, Db)

So each 5th we go down, the new key adds a flat (b).  In this order:-

#s: F+C+G+D+A+E+B  (Father Charles Goes Down And Ends Battle)

b's: Bb+Eb+Ab+Db+Gb+Cb+Fb   (Battle Ends And Down Goes Charles Father)

LEARN THIS. By just knowing how many # or b there are, you can use the cycle of 5ths to tell you which notes are # or b 

in each key, you can find the key signiature. (ie A has 3#s = F#,C#,G#)