We see that it's impossible to find the Keys of C and F Major using the Order of sharps and flats. These two Keys are exceptions to the rule. So we must Memorize their sharps and flats.
It's not hard, because the Key of C Major has NO sharps or flats:
And the Key of F Major has One flat:

Furthermore, the Scale Pattern of Steps for Major Keys remains the same:
So using these steps we can determine the Key Signatures for C and F Keys, even without knowing their flats or sharps. (Review Major Scale Steps.)
Also, since adding Keys' sharps or flats may become too numerous to remember, simply focus on those notes that are Natural (no sharps or flats).
The B Major Scale is:
It has 5 sharps! It's easier to remember that E and B notes are natural. Just play every note a half step up (sharp), except E and B.
We may wonder how it is possible to think this fast while playing? We must memorize the notes on our instrument, called fingerings. Though slow and challenging at first, we'll talk about practicing techniques. We'll understand what our instrument is doing in the Sound Waves page.
Order of Sharps:
An easy method to remember Sharps' order is:
Use the first letter of each word for the sharps' order.
Order of Flats:
To remember Flats' order, use:
...(spelling bead + GCF). Also, the flats' letters are the sharps' letters, in reverse order (backwards).
If we are able to find the name of the Key from the number of flats or sharps, why can't we do the reverse, and find the number of sharps or flats from just the Key's Name?
Sharps Example: If we only know that we are playing in the Key of A Major, take the A note, then move it down one half step, to find G#. Next, count the Order of sharps up to G#, like this:
Sharps example2: We only know we are playing in the Key of B Major. To find the last sharp, we must move B down a half step to A#. Then we follow the order of the sharps until we reach A#:
Now we know we have 5 sharps in the Key of B Major. (Why not remember that only E and B are natural?) There are 7 notes, 5 being sharp, and 2 being natural. Double the root, B natural:
In the Key of Ab Major, which flats do I play? We know that Ab is the second-to-last flat in the order of flats, for the key of Ab. Ordering flats until we go 1 flat past Ab is:
In Eb Major, we take the Eb, which will be the second-to-last flat. Then we follow the order of the flats to Eb, adding one more flat:
Some helpful reference points are the Sound of the scale and the order of the flats or sharps:
This is only the Order of flats and sharps, and not a Key.
The Sharp/Flat Rule page gives more in-depth information.
Scales have 7 notes, and we can double the first (root) note, to make 8 notes in a scale.
Within the CMaj scale,
...there are 2 C notes. The first is called the Root, and the last C is also the root, just 8 notes higher (counting the first C).
Octo is eight like an octopus. The notation for Octave in music is 8va, which says, "don't play that C, play the C above it". This applies to every scale letter.
Start any major scale, and we will end up an octave higher than where we started, landing on the root note again. This upper root is 8 scale steps from the bottom root, and has the same name (as in C to C, or F to F).
BUT, listen to the 1st and 8th notes... don't they sound like the same note when played together? This sound phenomena is one of the cornerstones of music. Most patterns in music revolve around two of the same notes, the higher and lower octaves. This Root grounds a song to a single tone, like a main idea.
The next part is important, to have the most freedom as a musician. We can create our own melodies, based on background chords (known as Improvisation).
The Major Scale has 8 notes, so use the C Major scale as an example:
Use the 1st, 3rd, and 5th notes (C E G), to create a Triad. These 3 notes Sound pleasant to the ear, and make up a Chord when played simultaneously.
This 1,3,5 pattern works in any scale. It gives the same chord even out of order, like 3,5,1 or 5,1,3 (called inversions). It will have a different 'feeling' if we play them in another order, but the chord's root (1) will always be the same (with its number of sharps or flats).
The root, or chord's name, ties the underlying sound together. The scale key signature will determine which notes are in a triad.
A Triad is built upon the root of a scale, so it will have the Key of the root, not the key of the 3rd or 5th notes. If we built a Triad upon either the 3rd or 5th notes, the Key would change to their root notes, and have sharps or flats that the C Triad does not have.
A different triad, from the F Major Scale:
The 1,3,5 pattern is FAC. Play them out of order (invert them), and we could have: ACF or CFA. But, we would still have the Key (#,b) of the Root(1), F.
This next part seems unrelated, but helps practice triads.
Let's change gears and learn the musical Staff! A staff is five horizontal Lines:

Staves are more than one Staff.
Let's populate this staff with letter notes, whether sharp, flat, or natural. Fill from the bottom line, starting with the letter 'E'. Notice that there are 4 Spaces between the Lines; so, 5 Lines, 4 Spaces.
Starting with 'E', begin the alphabet from there, and name the lines and spaces going up:

Nine letters fill 5 lines and 4 spaces, alphabetically.
Staff Lines starting from lowest to highest are EGBDF:

Remember it easily by using: Every Good Boy Does Fine.
Staff Spaces are FACE:

FACE is easy to remember, since it rhymes with 'space'. And remember, starting from 'E', the Lines and Spaces go up in alphabetical order:
Putting Triads in order on the staff shows the lowest to highest tones. When playing the tones together, individual notes are hard to distinguish.
Let's reference the E Major Scale:
Using the 1,3,5 pattern:
These notes match the first 3 Lines on the staff, EGB (forget sharps and flats for now, use naturals). Next, move up to the last note in the triad, B, and 1,3,5 from it, then repeat:
Placing them on the staff:

This helps us remember the pattern of the Triad. Repeat the letters a few times out loud, using the last letter of a triad for a new triad's root.
In the Staff section, starting from 'E', the lines and spaces are alphabetical (ascending the staff). We skip every other note to get a triad (1,3,5).
We remove the sharps and flats to focus on the Triads' letters. Memorizing the above triad pattern, our sharps and flats are faster to add.
Recommended reading: Tools of Sound